Content check — Elementary Algebra, Chapter 6: Polynomials

Every problem in the chapter beside the book’s answer, a chapter at a time as you scroll.

Chapter 6: Polynomials

622 problems · 0 figures

Be Prepared 6.1 §6.1 source ↗

Simplify:

8 x + 3 x .

If you missed this problem, review Example 1.24.

Example 1.24 in the book ↗

Answer

11 x

Be Prepared 6.2 §6.1 source ↗

Subtract:

(5 n + 8) -

(2 n - 1) .

If you missed this problem, review Example 1.139.

Example 1.139 in the book ↗

Answer

3 n + 9

Be Prepared 6.3 §6.1 source ↗

Write in expanded form:

a^{5} .

If you missed this problem, review Example 1.14.

Example 1.14 in the book ↗

Answer

a \cdot a \cdot a \cdot a \cdot a

Try It 6.1 §6.1 source ↗

Determine whether each polynomial is a monomial, binomial, trinomial, or other polynomial:

ⓐ $5 b$ ⓑ $8 y^{3} - 7 y^{2} - y - 3$ ⓒ $−3 x^{2} - 5 x + 9$ ⓓ $81 - 4 a^{2}$ ⓔ $−5 x^{6}$

Answer

ⓐ monomial ⓑ polynomial ⓒ trinomial ⓓ binomial ⓔ monomial

Try It 6.2 §6.1 source ↗

Determine whether each polynomial is a monomial, binomial, trinomial, or other polynomial:

ⓐ $27 z^{3} - 8$ ⓑ $12 m^{3} - 5 m^{2} - 2 m$ ⓒ $\frac{5}{6}$ ⓓ $8 x^{4} -$ $7 x^{2} -$ $6 x -$ $5$ ⓔ $− n^{4}$

Answer

ⓐ binomial ⓑ trinomial ⓒ monomial ⓓ polynomial ⓔ monomial

Try It 6.3 §6.1 source ↗

Find the degree of the following polynomials:

ⓐ $−15 b$ ⓑ $10 z^{4} + 4 z^{2} - 5$ ⓒ $12 c^{5} d^{4} +$ $9 c^{3} d^{9} -$ $7$ ⓓ $3 x^{2} y - 4 x$ ⓔ $−9$

Answer

ⓐ

1

ⓑ

4

ⓒ

12

ⓓ 3 ⓔ 0

Try It 6.4 §6.1 source ↗

Find the degree of the following polynomials:

ⓐ $52$ ⓑ $a^{4} b - 17 a^{4}$ ⓒ $5 x + 6 y + 2 z$ ⓓ $3 x^{2} - 5 x + 7$ ⓔ $− a^{3}$

Answer

ⓐ

0

ⓑ

5

ⓒ

1

ⓓ 2 ⓔ 3

Try It 6.5 §6.1 source ↗

Add:

12 q^{2} + 9 q^{2} .

Answer

21 q^{2}

Try It 6.6 §6.1 source ↗

Add:

−15 c^{2} + 8 c^{2} .

Answer

−7 c^{2}

Try It 6.7 §6.1 source ↗

Subtract:

8 m - (−5 m) .

Answer

13 m

Try It 6.8 §6.1 source ↗

Subtract:

−15 z^{3} -

(−5 z^{3}) .

Answer

−10 z^{3}

Try It 6.9 §6.1 source ↗

Add:

8 y^{2} +

3 z^{2} -

3 y^{2} .

Answer

5 y^{2} + 3 z^{2}

Try It 6.10 §6.1 source ↗

Add:

3 m^{2} + n^{2} - 7 m^{2} .

Answer

−4 m^{2} + n^{2}

Try It 6.11 §6.1 source ↗

Simplify:

m^{2} n^{2} -

8 m^{2} +

4 n^{2} .

Answer

There are no like terms to combine.

Try It 6.12 §6.1 source ↗

Simplify:

p q^{2} - 6 p - 5 q^{2} .

Answer

There are no like terms to combine.

Try It 6.13 §6.1 source ↗

Find the sum:

(7 x^{2} - 4 x + 5) +

(x^{2} - 7 x + 3) .

Answer

8 x^{2} - 11 x + 8

Try It 6.14 §6.1 source ↗

Find the sum:

(14 y^{2} + 6 y - 4) +

(3 y^{2} + 8 y + 5) .

Answer

17 y^{2} + 14 y + 1

Try It 6.15 §6.1 source ↗

Find the difference:

(8 x^{2} + 3 x - 19) -

(7 x^{2} - 14) .

Answer

x^{2} + 3 x - 5

Try It 6.16 §6.1 source ↗

Find the difference:

(9 b^{2} - 5 b - 4) -

(3 b^{2} - 5 b - 7) .

Answer

6 b^{2} + 3

Try It 6.17 §6.1 source ↗

Subtract:

(5 z^{2} - 6 z - 2)

from

(7 z^{2} + 6 z - 4)

Answer

2 z^{2} + 12 z - 2

Try It 6.18 §6.1 source ↗

Subtract:

(x^{2} - 5 x - 8)

from

(6 x^{2} + 9 x - 1)

Answer

5 x^{2} + 14 x + 7

Try It 6.19 §6.1 source ↗

Find the sum:

(3 x^{2} - 4 x y + 5 y^{2}) +

(2 x^{2} - x y)

Answer

5 x^{2} - 5 x y + 5 y^{2}

Try It 6.20 §6.1 source ↗

Find the sum:

(2 x^{2} - 3 x y - 2 y^{2}) +

(5 x^{2} - 3 x y)

Answer

7 x^{2} - 6 x y - 2 y^{2}

Try It 6.21 §6.1 source ↗

Find the difference:

(a^{2} + b^{2}) -

(a^{2} + 5 a b - 6 b^{2})

Answer

- 5 a b + 7 b^{2}

Try It 6.22 §6.1 source ↗

Find the difference:

(m^{2} + n^{2}) -

(m^{2} - 7 m n - 3 n^{2})

Answer

4 n^{2} + 7 m n

Try It 6.23 §6.1 source ↗

Simplify:

(x^{3} - x^{2} y) -

(x y^{2} + y^{3}) +

(x^{2} y + x y^{2})

Answer

x^{3} - y^{3}

Try It 6.24 §6.1 source ↗

Simplify:

(p^{3} - p^{2} q) +

(p q^{2} + q^{3}) -

(p^{2} q + p q^{2})

Answer

p^{3} - 2 p^{2} q + q^{3}

Try It 6.25 §6.1 source ↗

Evaluate: $3 x^{2} + 2 x - 15$ when

ⓐ $x = 3$

ⓑ $x = −5$

Answer

ⓐ

18

ⓑ

50

ⓒ

−15

Try It 6.26 §6.1 source ↗

Evaluate: $5 z^{2} - z - 4$ when

ⓐ $z = −2$

ⓑ $z = 0$

Answer

ⓐ

18

ⓑ

−4

ⓒ

14

Try It 6.27 §6.1 source ↗

The polynomial

−16 t^{2} + 250

gives the height of a ball

t

seconds after it is dropped from a 250-foot tall building. Find the height after

t = 0

seconds.

Answer

250

Try It 6.28 §6.1 source ↗

The polynomial

−16 t^{2} + 250

gives the height of a ball

t

seconds after it is dropped from a 250-foot tall building. Find the height after

t = 3

seconds.

Answer

106

Try It 6.29 §6.1 source ↗

The polynomial

6 x^{2} + 15 x y

gives the cost, in dollars, of producing a rectangular container whose top and bottom are squares with side x feet and sides of height y feet. Find the cost of producing a box with

x = 6

feet and

y = 4

feet.

Answer

$576

Try It 6.30 §6.1 source ↗

The polynomial

6 x^{2} + 15 x y

gives the cost, in dollars, of producing a rectangular container whose top and bottom are squares with side x feet and sides of height y feet. Find the cost of producing a box with

x = 5

feet and

y = 8

feet.

Answer

$750

6.1-1 §6.1 source ↗

Identify Polynomials, Monomials, Binomials, and Trinomials

In the following exercises, determine if each of the following polynomials is a monomial, binomial, trinomial, or other polynomial.

ⓐ $81 b^{5} - 24 b^{3} + 1$ ⓑ $5 c^{3} +$ $11 c^{2} -$ $c -$ $8$ ⓒ $\frac{14}{15} y + \frac{1}{7}$
ⓓ 5
ⓔ $4 y + 17$

Answer

ⓐ trinomial ⓑ polynomial ⓒ binomial ⓓ monomial ⓔ binomial

6.1-3 §6.1 source ↗

Identify Polynomials, Monomials, Binomials, and Trinomials

In the following exercises, determine if each of the following polynomials is a monomial, binomial, trinomial, or other polynomial.

ⓐ $8 - 3 x$ ⓑ $z^{2} - 5 z - 6$ ⓒ $y^{3} -$ $8 y^{2} +$ $2 y -$ $16$ ⓓ $81 b^{5} - 24 b^{3} + 1$ ⓔ $−18$

Answer

ⓐ binomial ⓑ trinomial ⓒ polynomial ⓓ trinomial ⓔ monomial

6.1-5 §6.1 source ↗

Determine the Degree of Polynomials

In the following exercises, determine the degree of each polynomial.

ⓐ $6 a^{2} + 12 a + 14$ ⓑ $18 x y^{2} z$ ⓒ $5 x + 2$ ⓓ $y^{3} -$ $8 y^{2} +$ $2 y -$ $16$ ⓔ $−24$

Answer

ⓐ 2 ⓑ 4 ⓒ 1 ⓓ 3 ⓔ 0

6.1-7 §6.1 source ↗

Determine the Degree of Polynomials

In the following exercises, determine the degree of each polynomial.

ⓐ $14 - 29 x$ ⓑ $z^{2} - 5 z - 6$ ⓒ $y^{3} -$ $8 y^{2} +$ $2 y -$ $16$ ⓓ $23 a b^{2} - 14$ ⓔ $−3$

Answer

ⓐ 1 ⓑ 2 ⓒ 3 ⓓ 3 ⓔ 0

6.1-9 §6.1 source ↗

Add and Subtract Monomials

In the following exercises, add or subtract the monomials.

${7x}^{2} + 5 x^{2}$

Answer

12 x^{2}

6.1-11 §6.1 source ↗

Add and Subtract Monomials

In the following exercises, add or subtract the monomials.

$−12 w + 18 w$

Answer

6 w

6.1-13 §6.1 source ↗

Add and Subtract Monomials

In the following exercises, add or subtract the monomials.

$4a - 9 a$

Answer

−5 a

6.1-15 §6.1 source ↗

Add and Subtract Monomials

In the following exercises, add or subtract the monomials.

$28 x - (−12 x)$

Answer

40 x

6.1-17 §6.1 source ↗

Add and Subtract Monomials

In the following exercises, add or subtract the monomials.

$−5 b - 17 b$

Answer

−22 b

6.1-19 §6.1 source ↗

Add and Subtract Monomials

In the following exercises, add or subtract the monomials.

$12 a + 5 b - 22 a$

Answer

−10a + 5 b

6.1-21 §6.1 source ↗

Add and Subtract Monomials

In the following exercises, add or subtract the monomials.

$2 a^{2} + b^{2} - 6 a^{2}$

Answer

−4 a^{2} + b^{2}

6.1-23 §6.1 source ↗

Add and Subtract Monomials

In the following exercises, add or subtract the monomials.

$x y^{2} - 5 x - 5 y^{2}$

Answer

x y^{2} - 5 x - 5 y^{2}

6.1-25 §6.1 source ↗

Add and Subtract Monomials

In the following exercises, add or subtract the monomials.

$a^{2} b - 4 a - 5 a b^{2}$

Answer

a^{2} b - 4 a - 5 a b^{2}

6.1-27 §6.1 source ↗

Add and Subtract Monomials

In the following exercises, add or subtract the monomials.

$12a + 8 b$

Answer

12a + 8 b

6.1-29 §6.1 source ↗

Add and Subtract Monomials

In the following exercises, add or subtract the monomials.

Add: $4 a, −3 b, −8 a$

Answer

−4 a - 3 b

6.1-31 §6.1 source ↗

Add and Subtract Monomials

In the following exercises, add or subtract the monomials.

Subtract $5 x^{6} from -$ $12 x^{6}$ .

Answer

−17 x^{6}

6.1-33 §6.1 source ↗

Add and Subtract Polynomials

In the following exercises, add or subtract the polynomials.

$(5 y^{2} + 12 y + 4) +$ $(6 y^{2} - 8 y + 7)$

Answer

11 y^{2} + 4 y + 11

6.1-35 §6.1 source ↗

Add and Subtract Polynomials

In the following exercises, add or subtract the polynomials.

$(x^{2} + 6 x + 8) +$ $(−4 x^{2} + 11 x - 9)$

Answer

−3 x^{2} + 17 x - 1

6.1-37 §6.1 source ↗

Add and Subtract Polynomials

In the following exercises, add or subtract the polynomials.

$(8 x^{2} - 5 x + 2) +$ $(3 x^{2} + 3)$

Answer

11 x^{2} - 5 x + 5

6.1-39 §6.1 source ↗

Add and Subtract Polynomials

In the following exercises, add or subtract the polynomials.

$(5 a^{2} + 8) +$ $(a^{2} - 4 a - 9)$

Answer

6 a^{2} - 4 a - 1

6.1-41 §6.1 source ↗

Add and Subtract Polynomials

In the following exercises, add or subtract the polynomials.

$(4 m^{2} - 6 m - 3) -$ $(2 m^{2} + m - 7)$

Answer

2 m^{2} - 7 m + 4

6.1-43 §6.1 source ↗

Add and Subtract Polynomials

In the following exercises, add or subtract the polynomials.

$(a^{2} + 8 a + 5) -$ $(a^{2} - 3 a + 2)$

Answer

11 a + 3

6.1-45 §6.1 source ↗

Add and Subtract Polynomials

In the following exercises, add or subtract the polynomials.

$(12 s^{2} - 15 s) -$ $(s - 9)$

Answer

12 s^{2} - 16 s + 9

6.1-47 §6.1 source ↗

Add and Subtract Polynomials

In the following exercises, add or subtract the polynomials.

Subtract $(9 x^{2} + 2)$ from $(12 x^{2} - x + 6)$ .

Answer

3 x^{2} - x + 4

6.1-49 §6.1 source ↗

Add and Subtract Polynomials

In the following exercises, add or subtract the polynomials.

Subtract $(7 w^{2} - 4 w + 2)$ from $(8 w^{2} - w + 6)$ .

Answer

w^{2} + 3 w + 4

6.1-51 §6.1 source ↗

Add and Subtract Polynomials

In the following exercises, add or subtract the polynomials.

Find the sum of $(2 p^{3} - 8)$ and $(p^{2} + 9 p + 18)$ .

Answer

2 p^{3} +

p^{2} +

9 p +

10

6.1-53 §6.1 source ↗

Add and Subtract Polynomials

In the following exercises, add or subtract the polynomials.

Find the sum of $(8 a^{3} - 8 a)$ and $(a^{2} + 6 a + 12)$ .

Answer

8 a^{3} +

a^{2} -

2 a +

12

6.1-55 §6.1 source ↗

Add and Subtract Polynomials

In the following exercises, add or subtract the polynomials.

Find the difference of
$(w^{2} + w - 42)$ and
$(w^{2} - 10 w + 24)$ .

Answer

11 w - 66

6.1-57 §6.1 source ↗

Add and Subtract Polynomials

In the following exercises, add or subtract the polynomials.

Find the difference of
$(c^{2} + 4 c - 33)$ and
$(c^{2} - 8 c + 12)$ .

Answer

12 c - 45

6.1-59 §6.1 source ↗

Add and Subtract Polynomials

In the following exercises, add or subtract the polynomials.

$(7 x^{2} - 2 x y + 6 y^{2}) +$ $(3 x^{2} - 5 x y)$

Answer

10 x^{2} -

7 x y +

6 y^{2}

6.1-61 §6.1 source ↗

Add and Subtract Polynomials

In the following exercises, add or subtract the polynomials.

$(7 m^{2} + m n - 8 n^{2}) +$ $(3 m^{2} + 2 m n)$

Answer

10 m^{2} +

3 m n -

8 n^{2}

6.1-63 §6.1 source ↗

Add and Subtract Polynomials

In the following exercises, add or subtract the polynomials.

$(a^{2} - b^{2}) -$ $(a^{2} + 3 a b - 4 b^{2})$

Answer

−3 a b + 3 b^{2}

6.1-65 §6.1 source ↗

Add and Subtract Polynomials

In the following exercises, add or subtract the polynomials.

$(u^{2} - v^{2}) -$ $(u^{2} - 4 u v - 3 v^{2})$

Answer

4 u v + 2 v^{2}

6.1-67 §6.1 source ↗

Add and Subtract Polynomials

In the following exercises, add or subtract the polynomials.

$(p^{3} - 3 p^{2} q) +$ $(2 p q^{2} + 4 q^{3}) -$ $(3 p^{2} q + p q^{2})$

Answer

p^{3} -

6 p^{2} q +

p q^{2} +

4 q^{3}

6.1-69 §6.1 source ↗

Add and Subtract Polynomials

In the following exercises, add or subtract the polynomials.

$(x^{3} - x^{2} y) -$ $(4 x y^{2} - y^{3}) +$ $(3 x^{2} y - x y^{2})$

Answer

x^{3} +

2 x^{2} y -

5 x y^{2} +

y^{3}

6.1-71 §6.1 source ↗

Evaluate a Polynomial for a Given Value

In the following exercises, evaluate each polynomial for the given value.

Evaluate $8 y^{2} - 3 y + 2$ when:

ⓐ $y = 5$ ⓑ $y = −2$ ⓒ $y = 0$

Answer

ⓐ 187 ⓑ 40 ⓒ 2

6.1-73 §6.1 source ↗

Evaluate a Polynomial for a Given Value

In the following exercises, evaluate each polynomial for the given value.

Evaluate $4 - 36 x$ when:

ⓐ $x = 3$ ⓑ $x = 0$ ⓒ $x = −1$

Answer

ⓐ −104 ⓑ 4 ⓒ 40

6.1-75 §6.1 source ↗

Evaluate a Polynomial for a Given Value

In the following exercises, evaluate each polynomial for the given value.

A painter drops a brush from a platform 75 feet high. The polynomial $−16 t^{2} + 75$ gives the height of the brush $t$ seconds after it was dropped. Find the height after $t = 2$ seconds.

Answer

6.1-77 §6.1 source ↗

Evaluate a Polynomial for a Given Value

In the following exercises, evaluate each polynomial for the given value.

A manufacturer of stereo sound speakers has found that the revenue received from selling the speakers at a cost of p dollars each is given by the polynomial $−4 p^{2} + 420 p .$ Find the revenue received when $p = 60$ dollars.

Answer

$10,800

6.1-79 §6.1 source ↗

Evaluate a Polynomial for a Given Value

In the following exercises, evaluate each polynomial for the given value.

Fuel Efficiency The fuel efficiency (in miles per gallon) of a car going at a speed of $x$ miles per hour is given by the polynomial $- \frac{1}{150} x^{2} + \frac{1}{3} x$ . Find the fuel efficiency when $x = 30 mph$ .

Answer

6.1-81 §6.1 source ↗

Evaluate a Polynomial for a Given Value

In the following exercises, evaluate each polynomial for the given value.

Rental Cost The cost to rent a rug cleaner for $d$ days is given by the polynomial $5.50 d + 25$ . Find the cost to rent the cleaner for 6 days.

Answer

$58

6.1-83 §6.1 source ↗

Evaluate a Polynomial for a Given Value

In the following exercises, evaluate each polynomial for the given value.

Temperature Conversion The temperature in degrees Fahrenheit is given by the polynomial $\frac{9}{5} c + 32$ where $c$ represents the temperature in degrees Celsius. Find the temperature in degrees Fahrenheit when $c = 65 ° .$

Answer

149

6.1-85 §6.1 source ↗

Evaluate a Polynomial for a Given Value

In the following exercises, evaluate each polynomial for the given value.

Using your own words, explain the difference between a polynomial with five terms and a polynomial with a degree of 5.

Answer

Answers will vary.

6.1-87 §6.1 source ↗

Evaluate a Polynomial for a Given Value

In the following exercises, evaluate each polynomial for the given value.

Jonathan thinks that $\frac{1}{3}$ and $\frac{1}{x}$ are both monomials. What is wrong with his reasoning?

Answer

Answers will vary.

Be Prepared 6.4 §6.2 source ↗

Simplify:

\frac{3}{4} \cdot \frac{3}{4} .

If you missed this problem, review Example 1.68.

Example 1.68 in the book ↗

Answer

\frac{3}{4} \cdot \frac{3}{4}

Be Prepared 6.5 §6.2 source ↗

Simplify:

(−2) (−2) (−2) .

If you missed this problem, review Example 1.50.

Example 1.50 in the book ↗

Answer

- 8

Try It 6.31 §6.2 source ↗

Simplify: ⓐ

6^{3}

ⓑ

15^{1}

ⓒ

{(\frac{3}{7})}^{2}

ⓓ

{(0.43)}^{2} .

Answer

ⓐ 216 ⓑ

15

ⓒ

\frac{9}{49}

ⓓ 0.1849

Try It 6.32 §6.2 source ↗

Simplify: ⓐ

2^{5}

ⓑ

21^{1}

ⓒ

{(\frac{2}{5})}^{3}

ⓓ

{(0.218)}^{2} .

Answer

ⓐ

32

ⓑ 21 ⓒ

\frac{8}{125}

ⓓ

0.047524

Try It 6.33 §6.2 source ↗

Simplify: ⓐ

{(−3)}^{4}

ⓑ

− 3^{4} .

Answer

ⓐ

81

ⓑ

−81

Try It 6.34 §6.2 source ↗

Simplify: ⓐ

{(−13)}^{2}

ⓑ

− 13^{2} .

Answer

ⓐ

169

ⓑ

−169

Try It 6.35 §6.2 source ↗

Simplify:

b^{9} \cdot b^{8} .

Answer

b^{17}

Try It 6.36 §6.2 source ↗

Simplify:

x^{12} \cdot x^{4} .

Answer

x^{16}

Try It 6.37 §6.2 source ↗

Simplify: ⓐ

5 \cdot 5^{5}

ⓑ

4^{9} \cdot 4^{9} .

Answer

ⓐ

5^{6}

ⓑ

4^{18}

Try It 6.38 §6.2 source ↗

Simplify: ⓐ

7^{6} \cdot 7^{8}

ⓑ

10 \cdot 10^{10} .

Answer

ⓐ

7^{14}

ⓑ

10^{11}

Try It 6.39 §6.2 source ↗

Simplify: ⓐ

p^{5} \cdot p

ⓑ

y^{14} \cdot y^{29} .

Answer

ⓐ

p^{6}

ⓑ

y^{43}

Try It 6.40 §6.2 source ↗

Simplify: ⓐ

z \cdot z^{7}

ⓑ

b^{15} \cdot b^{34} .

Answer

ⓐ

z^{8}

ⓑ

b^{49}

Try It 6.41 §6.2 source ↗

Simplify:

x^{6} \cdot x^{4} \cdot x^{8} .

Answer

x^{18}

Try It 6.42 §6.2 source ↗

Simplify:

b^{5} \cdot b^{9} \cdot b^{5} .

Answer

b^{19}

Try It 6.43 §6.2 source ↗

Simplify: ⓐ

{(b^{7})}^{5}

ⓑ

{(5^{4})}^{3} .

Answer

ⓐ

b^{35}

ⓑ

5^{12}

Try It 6.44 §6.2 source ↗

Simplify: ⓐ

{(z^{6})}^{9}

ⓑ

{(3^{7})}^{7} .

Answer

ⓐ

z^{54}

ⓑ

3^{49}

Try It 6.45 §6.2 source ↗

Simplify: ⓐ

{(−12 y)}^{2}

ⓑ

{(2 w x)}^{5} .

Answer

ⓐ

144 y^{2}

ⓑ

32 w^{5} x^{5}

Try It 6.46 §6.2 source ↗

Simplify: ⓐ

{(5 w x)}^{3}

ⓑ

{(−3 y)}^{3} .

Answer

ⓐ

125 w^{3} x^{3}

ⓑ

−27 y^{3}

Try It 6.47 §6.2 source ↗

Simplify: ⓐ

{(a^{4})}^{5} {(a^{7})}^{4}

ⓑ

{(−2 c^{4} d^{2})}^{3} .

Answer

ⓐ

a^{48}

ⓑ

−8 c^{12} d^{6}

Try It 6.48 §6.2 source ↗

Simplify: ⓐ

{(−3 x^{6} y^{7})}^{4}

ⓑ

{(q^{4})}^{5} {(q^{3})}^{3} .

Answer

ⓐ

81 x^{24} y^{28}

ⓑ

q^{29}

Try It 6.49 §6.2 source ↗

Simplify: ⓐ

{(5 n)}^{2} (3 n^{10})

ⓑ

{(c^{4} d^{2})}^{5} {(3 c d^{5})}^{4} .

Answer

ⓐ

75 n^{12}

ⓑ

81 c^{24} d^{30}

Try It 6.50 §6.2 source ↗

Simplify: ⓐ

{(a^{3} b^{2})}^{6} {(4 a b^{3})}^{4}

ⓑ

{(2 x)}^{3} (5 x^{7}) .

Answer

ⓐ

256 a^{22} b^{24}

ⓑ

40 x^{10}

Try It 6.51 §6.2 source ↗

Multiply:

(5 y^{7}) (−7 y^{4}) .

Answer

−35 y^{11}

Try It 6.52 §6.2 source ↗

Multiply:

(−6 b^{4}) (−9 b^{5}) .

Answer

54 b^{9}

Try It 6.53 §6.2 source ↗

Multiply:

(\frac{2}{5} a^{4} b^{3}) (15 a b^{3}) .

Answer

6 a^{5} b^{6}

Try It 6.54 §6.2 source ↗

Multiply:

(\frac{2}{3} r^{5} s) (12 r^{6} s^{7}) .

Answer

8 r^{11} s^{8}

6.2-89 §6.2 source ↗

Simplify Expressions with Exponents

In the following exercises, simplify each expression with exponents.

ⓐ $10^{4}$ ⓑ $17^{1}$ ⓒ ${(\frac{2}{9})}^{2}$ ⓓ ${(0.5)}^{3}$

Answer

ⓐ 10,000 ⓑ 17 ⓒ

\frac{4}{81}

ⓓ 0.125

6.2-91 §6.2 source ↗

Simplify Expressions with Exponents

In the following exercises, simplify each expression with exponents.

ⓐ $8^{3}$ ⓑ $8^{1}$ ⓒ ${(\frac{3}{4})}^{3}$ ⓓ ${(0.4)}^{3}$

Answer

ⓐ 512 ⓑ 8 ⓒ

\frac{27}{64}

ⓓ 0.064

6.2-93 §6.2 source ↗

Simplify Expressions with Exponents

In the following exercises, simplify each expression with exponents.

ⓐ ${(−2)}^{6}$ ⓑ $− 2^{6}$

Answer

ⓐ 64 ⓑ

−64

6.2-95 §6.2 source ↗

Simplify Expressions with Exponents

In the following exercises, simplify each expression with exponents.

ⓐ $− {(\frac{2}{3})}^{2}$ ⓑ ${(- \frac{2}{3})}^{2}$

Answer

ⓐ

- \frac{4}{9}

ⓑ

\frac{4}{9}

6.2-97 §6.2 source ↗

Simplify Expressions with Exponents

In the following exercises, simplify each expression with exponents.

ⓐ $− {0.1}^{4}$ ⓑ ${(−0.1)}^{4}$

Answer

ⓐ

−0.0001

ⓑ 0.0001

6.2-99 §6.2 source ↗

Simplify Expressions Using the Product Property for Exponents

In the following exercises, simplify each expression using the Product Property for Exponents.

$x^{4} \cdot x^{2}$

Answer

x^{6}

6.2-101 §6.2 source ↗

Simplify Expressions Using the Product Property for Exponents

In the following exercises, simplify each expression using the Product Property for Exponents.

$q^{27} \cdot q^{15}$

Answer

q^{42}

6.2-103 §6.2 source ↗

Simplify Expressions Using the Product Property for Exponents

In the following exercises, simplify each expression using the Product Property for Exponents.

ⓐ $3^{10} \cdot 3^{6}$ ⓑ $5 \cdot 5^{4}$

Answer

ⓐ

3^{16}

ⓑ

5^{5}

6.2-105 §6.2 source ↗

Simplify Expressions Using the Product Property for Exponents

In the following exercises, simplify each expression using the Product Property for Exponents.

ⓐ $w^{5} \cdot w^{}$ ⓑ $u^{41} \cdot u^{53}$

Answer

ⓐ

w^{6}

ⓑ

u^{94}

6.2-107 §6.2 source ↗

Simplify Expressions Using the Product Property for Exponents

In the following exercises, simplify each expression using the Product Property for Exponents.

$y \cdot y^{3} \cdot y^{5}$

Answer

y^{9}

6.2-109 §6.2 source ↗

Simplify Expressions Using the Product Property for Exponents

In the following exercises, simplify each expression using the Product Property for Exponents.

$c^{5} \cdot c^{11} \cdot c^{2}$

Answer

c^{18}

6.2-111 §6.2 source ↗

Simplify Expressions Using the Product Property for Exponents

In the following exercises, simplify each expression using the Product Property for Exponents.

$n^{y} \cdot n^{2}$

Answer

n^{y + 2}

6.2-113 §6.2 source ↗

Simplify Expressions Using the Product Property for Exponents

In the following exercises, simplify each expression using the Product Property for Exponents.

$x^{p} \cdot x^{q}$

Answer

x^{p + q}

6.2-115 §6.2 source ↗

Simplify Expressions Using the Power Property for Exponents

In the following exercises, simplify each expression using the Power Property for Exponents.

ⓐ ${(b^{2})}^{7}$ ⓑ ${(3^{8})}^{2}$

Answer

ⓐ

b^{14}

ⓑ

3^{16}

6.2-117 §6.2 source ↗

Simplify Expressions Using the Power Property for Exponents

In the following exercises, simplify each expression using the Power Property for Exponents.

ⓐ ${(x^{2})}^{y}$ ⓑ ${(7^{a})}^{b}$

Answer

ⓐ

x^{2 y}

ⓑ

7^{a b}

6.2-119 §6.2 source ↗

Simplify Expressions Using the Product to a Power Property

In the following exercises, simplify each expression using the Product to a Power Property.

ⓐ ${(5 x)}^{2}$ ⓑ ${(4 a b)}^{2}$

Answer

ⓐ

25 x^{2}

ⓑ

16 a^{2} b^{2}

6.2-121 §6.2 source ↗

Simplify Expressions Using the Product to a Power Property

In the following exercises, simplify each expression using the Product to a Power Property.

ⓐ ${(−7 n)}^{3}$ ⓑ ${(3 x y z)}^{4}$

Answer

ⓐ

−343 n^{3}

ⓑ

81 x^{4} y^{4} z^{4}

6.2-123 §6.2 source ↗

Simplify Expressions by Applying Several Properties

In the following exercises, simplify each expression.

ⓐ ${(w^{4})}^{3} \cdot {(w^{5})}^{2}$ ⓑ ${(2 x y^{4})}^{5}$

Answer

ⓐ

w^{22}

ⓑ

32 x^{5} y^{20}

6.2-125 §6.2 source ↗

Simplify Expressions by Applying Several Properties

In the following exercises, simplify each expression.

ⓐ ${(−10 q^{2} p^{4})}^{3}$ ⓑ ${(n^{3})}^{10} \cdot {(n^{5})}^{2}$

Answer

ⓐ

−1000 q^{6} p^{12}

ⓑ

n^{40}

6.2-127 §6.2 source ↗

Simplify Expressions by Applying Several Properties

In the following exercises, simplify each expression.

ⓐ ${(2 y)}^{3} (6 y)$ ⓑ ${(10 k^{4})}^{3} {(5 k^{6})}^{2}$

Answer

ⓐ

48 y^{4}

ⓑ

25,000 k^{24}

6.2-129 §6.2 source ↗

Simplify Expressions by Applying Several Properties

In the following exercises, simplify each expression.

ⓐ ${(4 b)}^{2} {(3 b)}^{3}$ ⓑ ${(\frac{1}{2} j^{2})}^{5} {(\frac{2}{5} j^{3})}^{2}$

Answer

ⓐ

432 b^{5}

ⓑ

\frac{1}{200} j^{16}

6.2-131 §6.2 source ↗

Simplify Expressions by Applying Several Properties

In the following exercises, simplify each expression.

ⓐ ${(2 r^{2})}^{3} {(4 r)}^{2}$ ⓑ ${(3 x^{3})}^{3} {(x^{5})}^{4}$

Answer

ⓐ

128 r^{8}

ⓑ

{27 x}^{29}

6.2-133 §6.2 source ↗

Multiply Monomials

In the following exercises, multiply the monomials.

$(6 y^{7}) (−3 y^{4})$

Answer

−18 y^{11}

6.2-135 §6.2 source ↗

Multiply Monomials

In the following exercises, multiply the monomials.

$(−8 u^{6}) (−9 u)$

Answer

72 u^{7}

6.2-137 §6.2 source ↗

Multiply Monomials

In the following exercises, multiply the monomials.

$(\frac{1}{5} f^{8}) (20 f^{3})$

Answer

4 f^{11}

6.2-139 §6.2 source ↗

Multiply Monomials

In the following exercises, multiply the monomials.

$(4 a^{3} b) (9 a^{2} b^{6})$

Answer

36 a^{5} b^{7}

6.2-141 §6.2 source ↗

Multiply Monomials

In the following exercises, multiply the monomials.

$(\frac{4}{7} r s^{2}) (14 r s^{3})$

Answer

8 r^{2} s^{5}

6.2-143 §6.2 source ↗

Multiply Monomials

In the following exercises, multiply the monomials.

$(\frac{2}{3} x^{2} y) (\frac{3}{4} x y^{2})$

Answer

\frac{1}{2} x^{3} y^{3}

6.2-145 §6.2 source ↗

Mixed Practice

In the following exercises, simplify each expression.

${(x^{2})}^{4} \cdot {(x^{3})}^{2}$

Answer

x^{14}

6.2-147 §6.2 source ↗

Mixed Practice

In the following exercises, simplify each expression.

${(a^{2})}^{6} \cdot {(a^{3})}^{8}$

Answer

a^{36}

6.2-149 §6.2 source ↗

Mixed Practice

In the following exercises, simplify each expression.

${(2 m^{6})}^{3}$

Answer

8 m^{18}

6.2-151 §6.2 source ↗

Mixed Practice

In the following exercises, simplify each expression.

${(10 x^{2} y)}^{3}$

Answer

1000 x^{6} y^{3}

6.2-153 §6.2 source ↗

Mixed Practice

In the following exercises, simplify each expression.

${(−2 a^{3} b^{2})}^{4}$

Answer

16 a^{12} b^{8}

6.2-155 §6.2 source ↗

Mixed Practice

In the following exercises, simplify each expression.

${(\frac{2}{3} x^{2} y)}^{3}$

Answer

\frac{8}{27} x^{6} y^{3}

6.2-157 §6.2 source ↗

Mixed Practice

In the following exercises, simplify each expression.

${(8 a^{3})}^{2} {(2 a)}^{4}$

Answer

1024 a^{10}

6.2-159 §6.2 source ↗

Mixed Practice

In the following exercises, simplify each expression.

${(10 p^{4})}^{3} {(5 p^{6})}^{2}$

Answer

25000 p^{24}

6.2-161 §6.2 source ↗

Mixed Practice

In the following exercises, simplify each expression.

${(\frac{1}{2} x^{2} y^{3})}^{4} {(4 x^{5} y^{3})}^{2}$

Answer

x^{18} y^{18}

6.2-163 §6.2 source ↗

Mixed Practice

In the following exercises, simplify each expression.

${(3 m^{2} n)}^{2} {(2 m n^{5})}^{4}$

Answer

144 m^{8} n^{22}

6.2-165 §6.2 source ↗

Mixed Practice

In the following exercises, simplify each expression.

Email Kate emails a flyer to ten of her friends and tells them to forward it to ten of their friends, who forward it to ten of their friends, and so on. The number of people who receive the email on the second round is $10^{2}$ , on the third round is $10^{3}$ , as shown in the table below. How many people will receive the email on the sixth round? Simplify the expression to show the number of people who receive the email.

Round	Number of people
1	10
2	$10^{2}$
3	$10^{3}$
…	…
6	?

Answer

1,000,000

6.2-167 §6.2 source ↗

Mixed Practice

In the following exercises, simplify each expression.

Clearance A department store is clearing out merchandise in order to make room for new inventory. The plan is to mark down items by 30% each week. This means that each week the cost of an item is 70% of the previous week’s cost. If the original cost of a sofa was $1,000, the cost for the first week would be $$ 1,000 (0.70)$ and the cost of the item during the second week would be $$ 1,000 {(0.70)}^{2}$ . Complete the table shown below. What will be the cost of the sofa during the fifth week? Simplify the expression, to show the cost in dollars.

Week	Cost
1	$$ 1,000 (0.70)$
2	$$ 1,000 {(0.70)}^{2}$
3
4	…
5	?

Answer

$168.07

6.2-169 §6.2 source ↗

Mixed Practice

In the following exercises, simplify each expression.

Use the Product Property for Exponents to explain why $x \cdot x = x^{2}$ .

Answer

Answers will vary.

6.2-171 §6.2 source ↗

Mixed Practice

In the following exercises, simplify each expression.

Jorge thinks ${(\frac{1}{2})}^{2}$ is 1. What is wrong with his reasoning?

Answer

Answers will vary.

Be Prepared 6.6 §6.3 source ↗

Distribute:

2 (x + 3) .

If you missed this problem, review Example 1.132.

Example 1.132 in the book ↗

Answer

2 x + 6

Be Prepared 6.7 §6.3 source ↗

Combine like terms:

x^{2} +

9 x +

7 x +

63 .

If you missed this problem, review Example 1.24.

Example 1.24 in the book ↗

Answer

x^{2} + 16 x + 63

Try It 6.55 §6.3 source ↗

Multiply:

5 (x + 7) .

Answer

5 x + 35

Try It 6.56 §6.3 source ↗

Multiply:

3 (y + 13) .

Answer

3 y + 39

Try It 6.57 §6.3 source ↗

Multiply:

x (x - 7) .

Answer

x^{2} - 7 x

Try It 6.58 §6.3 source ↗

Multiply:

d (d - 11) .

Answer

d^{2} - 11 d

Try It 6.59 §6.3 source ↗

Multiply:

5 x (x + 4 y) .

Answer

5 x^{2} + 20 x y

Try It 6.60 §6.3 source ↗

Multiply:

2 p (6 p + r) .

Answer

12 p^{2} + 2 p r

Try It 6.61 §6.3 source ↗

Multiply:

−3 y (5 y^{2} + 8 y - 7) .

Answer

−15 y^{3} -

24 y^{2} +

21 y

Try It 6.62 §6.3 source ↗

Multiply:

4 x^{2} (2 x^{2} - 3 x + 5) .

Answer

8 x^{4} -

12 x^{3} +

20 x^{2}

Try It 6.63 §6.3 source ↗

Multiply:

4 x (3 x^{2} - 5 x + 3) .

Answer

12 x^{3} -

20 x^{2} +

12 x

Try It 6.64 §6.3 source ↗

Multiply:

−6 a^{3} (3 a^{2} - 2 a + 6) .

Answer

−18 a^{5} +

12 a^{4} -

36 a^{3}

Try It 6.65 §6.3 source ↗

Multiply:

(x + 8) p .

Answer

x p + 8 p

Try It 6.66 §6.3 source ↗

Multiply:

(a + 4) p .

Answer

a p + 4 p

Try It 6.67 §6.3 source ↗

Multiply:

(x + 8) (x + 9) .

Answer

x^{2} + 17 x + 72

Try It 6.68 §6.3 source ↗

Multiply:

(5 x + 9) (4 x + 3) .

Answer

20 x^{2} + 51 x + 27

Try It 6.69 §6.3 source ↗

Multiply:

(3 b + 5) (4 b + 6) .

Answer

12 b^{2} + 38 b + 30

Try It 6.70 §6.3 source ↗

Multiply:

(a + 10) (a + 7) .

Answer

a^{2} + 17 a + 70

Try It 6.71 §6.3 source ↗

Multiply:

(5 y + 2) (6 y - 3) .

Answer

30 y^{2} - 3 y - 6

Try It 6.72 §6.3 source ↗

Multiply:

(3 c + 4) (5 c - 2) .

Answer

15 c^{2} + 14 c - 8

Try It 6.73 §6.3 source ↗

Multiply:

(a + 7) (a - b) .

Answer

a^{2} - a b + 7 a - 7 b

Try It 6.74 §6.3 source ↗

Multiply:

(x + 5) (x - y) .

Answer

x^{2} - x y + 5 x - 5 y

Try It 6.75 §6.3 source ↗

Multiply using the FOIL method:

(x + 6) (x + 8) .

Answer

x^{2} + 14 x + 48

Try It 6.76 §6.3 source ↗

Multiply using the FOIL method:

(y + 17) (y + 3) .

Answer

y^{2} + 20 y + 51

Try It 6.77 §6.3 source ↗

Multiply:

(x - 7) (x + 5) .

Answer

x^{2} - 2 x - 35

Try It 6.78 §6.3 source ↗

Multiply:

(b - 3) (b + 6) .

Answer

b^{2} + 3 b - 18

Try It 6.79 §6.3 source ↗

Multiply:

(3 x + 7) (5 x - 2) .

Answer

15 x^{2} + 29 x - 14

Try It 6.80 §6.3 source ↗

Multiply:

(4 y + 5) (4 y - 10) .

Answer

16 y^{2} - 20 y - 50

Try It 6.81 §6.3 source ↗

Multiply:

(10 c - d) (c - 6) .

Answer

10 c^{2} -

60 c -

c d +

6 d

Try It 6.82 §6.3 source ↗

Multiply:

(7 x - y) (2 x - 5) .

Answer

14 x^{2} -

35 x -

2 x y +

5 y

Try It 6.83 §6.3 source ↗

Multiply:

(x^{2} + 6) (x - 8) .

Answer

x^{3} -

8 x^{2} +

6 x -

48

Try It 6.84 §6.3 source ↗

Multiply:

(y^{2} + 7) (y - 9) .

Answer

y^{3} -

9 y^{2} +

7 y -

63

Try It 6.85 §6.3 source ↗

Multiply:

(2 a b + 5) (4 a b - 4) .

Answer

8 a^{2} b^{2} +

12 a b -

20

Try It 6.86 §6.3 source ↗

Multiply:

(2 x y + 3) (4 x y - 5) .

Answer

8 x^{2} y^{2} +

2 x y -

15

Try It 6.87 §6.3 source ↗

Multiply using the Vertical Method:

(5 m - 7) (3 m - 6) .

Answer

15 m^{2} - 51 m + 42

Try It 6.88 §6.3 source ↗

Multiply using the Vertical Method:

(6 b - 5) (7 b - 3) .

Answer

42 b^{2} - 53 b + 15

Try It 6.89 §6.3 source ↗

Multiply using the Distributive Property:

(y - 3) (y^{2} - 5 y + 2) .

Answer

y^{3} -

8 y^{2} +

17 y -

6

Try It 6.90 §6.3 source ↗

Multiply using the Distributive Property:

(x + 4) (2 x^{2} - 3 x + 5) .

Answer

2 x^{3} +

5 x^{2} -

7 x +

20

Try It 6.91 §6.3 source ↗

Multiply using the Vertical Method:

(y - 3) (y^{2} - 5 y + 2) .

Answer

y^{3} -

8 y^{2} +

17 y -

6

Try It 6.92 §6.3 source ↗

Multiply using the Vertical Method:

(x + 4) (2 x^{2} - 3 x + 5) .

Answer

2 x^{3} +

5 x^{2} -

7 x +

20

6.3-173 §6.3 source ↗

Multiply a Polynomial by a Monomial

In the following exercises, multiply.

$4 (w + 10)$

Answer

4 w + 40

6.3-175 §6.3 source ↗

Multiply a Polynomial by a Monomial

In the following exercises, multiply.

$−3 (a + 7)$

Answer

−3 a - 21

6.3-177 §6.3 source ↗

Multiply a Polynomial by a Monomial

In the following exercises, multiply.

$2 (x - 7)$

Answer

2 x - 14

6.3-179 §6.3 source ↗

Multiply a Polynomial by a Monomial

In the following exercises, multiply.

$−3 (k - 4)$

Answer

−3 k + 12

6.3-181 §6.3 source ↗

Multiply a Polynomial by a Monomial

In the following exercises, multiply.

$q (q + 5)$

Answer

q^{2} + 5 q

6.3-183 §6.3 source ↗

Multiply a Polynomial by a Monomial

In the following exercises, multiply.

$− b (b + 9)$

Answer

− b^{2} - 9 b

6.3-185 §6.3 source ↗

Multiply a Polynomial by a Monomial

In the following exercises, multiply.

$− x (x - 10)$

Answer

− x^{2} + 10 x

6.3-187 §6.3 source ↗

Multiply a Polynomial by a Monomial

In the following exercises, multiply.

$6 r (4 r + s)$

Answer

24 r^{2} + 6 r s

6.3-189 §6.3 source ↗

Multiply a Polynomial by a Monomial

In the following exercises, multiply.

$12 x (x - 10)$

Answer

12 x^{2} - 120 x

6.3-191 §6.3 source ↗

Multiply a Polynomial by a Monomial

In the following exercises, multiply.

$−9 a (3 a + 5)$

Answer

−27 a^{2} - 45 a

6.3-193 §6.3 source ↗

Multiply a Polynomial by a Monomial

In the following exercises, multiply.

$3 (p^{2} + 10 p + 25)$

Answer

3 p^{2} + 30 p + 75

6.3-195 §6.3 source ↗

Multiply a Polynomial by a Monomial

In the following exercises, multiply.

$−8 x (x^{2} + 2 x - 15)$

Answer

−8 x^{3} -

16 x^{2} +

120 x

6.3-197 §6.3 source ↗

Multiply a Polynomial by a Monomial

In the following exercises, multiply.

$5 q^{3} (q^{3} - 2 q + 6)$

Answer

5 q^{6} -

10 q^{4} +

30 q^{3}

6.3-199 §6.3 source ↗

Multiply a Polynomial by a Monomial

In the following exercises, multiply.

$−8 y (y^{2} + 2 y - 15)$

Answer

−8 y^{3} -

16 y^{2} +

120 y

6.3-201 §6.3 source ↗

Multiply a Polynomial by a Monomial

In the following exercises, multiply.

$5 q^{3} (q^{2} - 2 q + 6)$

Answer

5 q^{5} -

10 q^{4} +

30 q^{3}

6.3-203 §6.3 source ↗

Multiply a Polynomial by a Monomial

In the following exercises, multiply.

$−4 z^{2} (3 z^{2} + 12 z - 1)$

Answer

−12 z^{4} -

48 z^{3} +

4 z^{2}

6.3-205 §6.3 source ↗

Multiply a Polynomial by a Monomial

In the following exercises, multiply.

$(2 m - 9) m$

Answer

2 m^{2} - 9 m

6.3-207 §6.3 source ↗

Multiply a Polynomial by a Monomial

In the following exercises, multiply.

$(w - 6) \cdot 8$

Answer

8 w - 48

6.3-209 §6.3 source ↗

Multiply a Polynomial by a Monomial

In the following exercises, multiply.

$4 (x + 10)$

Answer

4 x + 40

6.3-211 §6.3 source ↗

Multiply a Polynomial by a Monomial

In the following exercises, multiply.

$15 (r - 24)$

Answer

15 r - 360

6.3-213 §6.3 source ↗

Multiply a Polynomial by a Monomial

In the following exercises, multiply.

$−3 (m + 11)$

Answer

−3 m - 33

6.3-215 §6.3 source ↗

Multiply a Polynomial by a Monomial

In the following exercises, multiply.

$−8 (z - 5)$

Answer

−8 z + 40

6.3-217 §6.3 source ↗

Multiply a Polynomial by a Monomial

In the following exercises, multiply.

$u (u + 5)$

Answer

u^{2} + 5 u

6.3-219 §6.3 source ↗

Multiply a Polynomial by a Monomial

In the following exercises, multiply.

$n (n^{2} - 3 n)$

Answer

n^{3} - 3 n^{2}

6.3-221 §6.3 source ↗

Multiply a Polynomial by a Monomial

In the following exercises, multiply.

$6 x (4 x + y)$

Answer

24 x^{2} + 6 x y

6.3-223 §6.3 source ↗

Multiply a Polynomial by a Monomial

In the following exercises, multiply.

$5 p (11 p - 5 q)$

Answer

55 p^{2} - 25 p q

6.3-225 §6.3 source ↗

Multiply a Polynomial by a Monomial

In the following exercises, multiply.

$3 (v^{2} + 10 v + 25)$

Answer

3 v^{2} + 30 v + 75

6.3-227 §6.3 source ↗

Multiply a Polynomial by a Monomial

In the following exercises, multiply.

$2 n (4 n^{2} - 4 n + 1)$

Answer

8 n^{3} - 8 n^{2} + 2 n

6.3-229 §6.3 source ↗

Multiply a Polynomial by a Monomial

In the following exercises, multiply.

$−8 y (y^{2} + 2 y - 15)$

Answer

−8 y^{3} -

16 y^{2} +

120 y

6.3-231 §6.3 source ↗

Multiply a Polynomial by a Monomial

In the following exercises, multiply.

$5 q^{3} (q^{2} - 2 q + 6)$

Answer

5 q^{5} -

10 q^{4} +

30 q^{3}

6.3-233 §6.3 source ↗

Multiply a Polynomial by a Monomial

In the following exercises, multiply.

$−4 z^{2} (3 z^{2} + 12 z - 1)$

Answer

−12 z^{4} -

48 z^{3} +

4 z^{2}

6.3-235 §6.3 source ↗

Multiply a Polynomial by a Monomial

In the following exercises, multiply.

$(2 y - 9) y$

Answer

2 y^{2} - 9 y

6.3-237 §6.3 source ↗

Multiply a Binomial by a Binomial

In the following exercises, multiply the following binomials using: ⓐ the Distributive Property ⓑ the FOIL method ⓒ the Vertical Method.

$(w + 5) (w + 7)$

Answer

w^{2} + 12 w + 35

6.3-239 §6.3 source ↗

Multiply a Binomial by a Binomial

In the following exercises, multiply the following binomials using: ⓐ the Distributive Property ⓑ the FOIL method ⓒ the Vertical Method.

$(p + 11) (p - 4)$

Answer

p^{2} + 7 p - 44

6.3-241 §6.3 source ↗

In the following exercises, multiply the binomials. Use any method.

$(x + 8) (x + 3)$

Answer

x^{2} + 11 x + 24

6.3-243 §6.3 source ↗

In the following exercises, multiply the binomials. Use any method.

$(y - 6) (y - 2)$

Answer

y^{2} - 8 y + 12

6.3-245 §6.3 source ↗

In the following exercises, multiply the binomials. Use any method.

$(w - 4) (w + 7)$

Answer

w^{2} + 3 w - 28

6.3-247 §6.3 source ↗

In the following exercises, multiply the binomials. Use any method.

$(p + 12) (p - 5)$

Answer

p^{2} + 7 p - 60

6.3-249 §6.3 source ↗

In the following exercises, multiply the binomials. Use any method.

$(6 p + 5) (p + 1)$

Answer

6 p^{2} + 11 p + 5

6.3-251 §6.3 source ↗

In the following exercises, multiply the binomials. Use any method.

$(2 t - 9) (10 t + 1)$

Answer

20 t^{2} - 88 t - 9

6.3-253 §6.3 source ↗

In the following exercises, multiply the binomials. Use any method.

$(5 x - y) (3 x - 6)$

Answer

15 x^{2} -

3 x y -

30 x +

6 y

6.3-255 §6.3 source ↗

In the following exercises, multiply the binomials. Use any method.

$(a + b) (2 a + 3 b)$

Answer

2 a^{2} + 5 a b + 3 b^{2}

6.3-257 §6.3 source ↗

In the following exercises, multiply the binomials. Use any method.

$(4 z - y) (z - 6)$

Answer

4 z^{2} -

24 z -

z y +

6 y

6.3-259 §6.3 source ↗

In the following exercises, multiply the binomials. Use any method.

$(x^{2} + 3) (x + 2)$

Answer

x^{3} + 2 x^{2} + 3 x + 6

6.3-261 §6.3 source ↗

In the following exercises, multiply the binomials. Use any method.

$(x^{2} + 8) (x^{2} - 5)$

Answer

x^{4} + 3 x^{2} - 40

6.3-263 §6.3 source ↗

In the following exercises, multiply the binomials. Use any method.

$(5 a b - 1) (2 a b + 3)$

Answer

10 a^{2} b^{2} +

13 a b -

3

6.3-265 §6.3 source ↗

In the following exercises, multiply the binomials. Use any method.

$(6 p q - 3) (4 p q - 5)$

Answer

24 p^{2} q^{2} -

42 p q +

15

6.3-267 §6.3 source ↗

Multiply a Trinomial by a Binomial

In the following exercises, multiply using ⓐ the Distributive Property ⓑ the Vertical Method.

$(x + 5) (x^{2} + 4 x + 3)$

Answer

x^{3} +

9 x^{2} +

23 x +

15

6.3-269 §6.3 source ↗

Multiply a Trinomial by a Binomial

In the following exercises, multiply using ⓐ the Distributive Property ⓑ the Vertical Method.

$(y + 8) (4 y^{2} + y - 7)$

Answer

4 y^{3} +

33 y^{2} +

y -

56

6.3-271 §6.3 source ↗

In the following exercises, multiply. Use either method.

$(w - 7) (w^{2} - 9 w + 10)$

Answer

w^{3} -

16 w^{2} +

73 w -

70

6.3-273 §6.3 source ↗

In the following exercises, multiply. Use either method.

$(3 q + 1) (q^{2} - 4 q - 5)$

Answer

3 q^{3} -

11 q^{2} -

19 q -

5

6.3-275 §6.3 source ↗

Mixed Practice

$(10 y - 6) +$ $(4 y - 7)$

Answer

14 y - 13

6.3-277 §6.3 source ↗

Mixed Practice

$(x^{2} - 4 x - 34) -$ $(x^{2} + 7 x - 6)$

Answer

−11 x - 28

6.3-279 §6.3 source ↗

Mixed Practice

$5 q (3 q^{2} - 6 q + 11)$

Answer

15 q^{3} -

30 q^{2} +

55 q

6.3-281 §6.3 source ↗

Mixed Practice

$(s - 7) (s + 9)$

Answer

s^{2} + 2 s - 63

6.3-283 §6.3 source ↗

Mixed Practice

$(y^{2} - 2 y) (y + 1)$

Answer

y^{3} - y^{2} - 2 y

6.3-285 §6.3 source ↗

Mixed Practice

$(3 n - 4) (n^{2} + n - 7)$

Answer

3 n^{3} -

n^{2} -

25 n +

28

6.3-287 §6.3 source ↗

Mixed Practice

$(7 p + 10) (7 p - 10)$

Answer

49 p^{2} - 100

6.3-289 §6.3 source ↗

Mixed Practice

$(4 m^{2} - 3 m - 7) m^{2}$

Answer

4 m^{4} - 3 m^{3} - 7 m^{2}

6.3-291 §6.3 source ↗

Mixed Practice

$(5 a + 7 b) (5 a + 7 b)$

Answer

25 a^{2} +

70 a b +

49 b^{2}

6.3-293 §6.3 source ↗

Mixed Practice

$(4 y + 12 z) (4 y - 12 z)$

Answer

16 y^{2} - 144 z^{2}

6.3-295 §6.3 source ↗

Mixed Practice

Mental math You can use binomial multiplication to multiply numbers without a calculator. Say you need to multiply 18 times 17. Think of 18 as $20 - 2$ and 17 as $20 - 3$ .

ⓐ Multiply $(20 - 2) (20 - 3)$ by the FOIL method.

ⓑ Multiply $18 \cdot 17$ without using a calculator.

Answer

ⓐ 306 ⓑ 306 ⓒ Answers will vary.

6.3-297 §6.3 source ↗

Mixed Practice

Which method do you prefer to use when multiplying a trinomial by a binomial: the Distributive Property or the Vertical Method? Why?

Answer

Answers will vary.

6.3-299 §6.3 source ↗

Mixed Practice

Multiply the following:

$\begin{matrix} (m - 3) (m + 3) \\ (n - 10) (n + 10) \\ (p - 8) (p + 8) \end{matrix}$

Explain the pattern that you see in your answers.

Answer

Answers may vary.

6.3-301 §6.3 source ↗

Mixed Practice

Multiply the following:

$\begin{matrix} (x - 4) (x - 4) \\ (y - 1) (y - 1) \\ (z - 7) (z - 7) \end{matrix}$

Explain the pattern that you see in your answers.

Answer

Answers may vary.

Be Prepared 6.8 §6.4 source ↗

Simplify: ⓐ

9^{2}

ⓑ

{(−9)}^{2}

ⓒ

− 9^{2} .

If you missed this problem, review Example 1.50.

Example 1.50 in the book ↗

Answer

ⓐ

81

ⓑ

81

ⓒ

- 81

Try It 6.93 §6.4 source ↗

Multiply:

{(x + 9)}^{2} .

Answer

x^{2} + 18 x + 81

Try It 6.94 §6.4 source ↗

Multiply:

{(y + 11)}^{2} .

Answer

y^{2} + 22 y + 121

Try It 6.95 §6.4 source ↗

Multiply:

{(x - 9)}^{2} .

Answer

x^{2} - 18 x + 81

Try It 6.96 §6.4 source ↗

Multiply:

{(p - 13)}^{2} .

Answer

p^{2} - 26 p + 169

Try It 6.97 §6.4 source ↗

Multiply:

{(6 x + 3)}^{2} .

Answer

36 x^{2} + 36 x + 9

Try It 6.98 §6.4 source ↗

Multiply:

{(4 x + 9)}^{2} .

Answer

16 x^{2} + 72 x + 81

Try It 6.99 §6.4 source ↗

Multiply:

{(2 c - d)}^{2} .

Answer

4 c^{2} - 4 c d + d^{2}

Try It 6.100 §6.4 source ↗

Multiply:

{(4 x - 5 y)}^{2} .

Answer

16 x^{2} -

40 x y +

25 y^{2}

Try It 6.101 §6.4 source ↗

Multiply:

{(2 x^{2} + 1)}^{2} .

Answer

4 x^{4} + 4 x^{2} + 1

Try It 6.102 §6.4 source ↗

Multiply:

{(3 y^{3} + 2)}^{2} .

Answer

9 y^{6} + 12 y^{3} + 4

Try It 6.103 §6.4 source ↗

Multiply:

(x - 5) (x + 5) .

Answer

x^{2} - 25

Try It 6.104 §6.4 source ↗

Multiply:

(w - 3) (w + 3) .

Answer

w^{2} - 9

Try It 6.105 §6.4 source ↗

Multiply:

(6 x + 5) (6 x - 5) .

Answer

36 x^{2} - 25

Try It 6.106 §6.4 source ↗

Multiply:

(2 x + 7) (2 x - 7) .

Answer

4 x^{2} - 49

Try It 6.107 §6.4 source ↗

Multiply:

(7 + 4 x) (7 - 4 x) .

Answer

49 - 16 x^{2}

Try It 6.108 §6.4 source ↗

Multiply:

(9 - 2 y) (9 + 2 y) .

Answer

81 - 4 y^{2}

Try It 6.109 §6.4 source ↗

Find the product:

(4 p - 7 q) (4 p + 7 q) .

Answer

16 p^{2} - 49 q^{2}

Try It 6.110 §6.4 source ↗

Find the product:

(3 x - y) (3 x + y) .

Answer

9 x^{2} - y^{2}

Try It 6.111 §6.4 source ↗

Find the product:

(x y - 6) (x y + 6) .

Answer

x^{2} y^{2} - 36

Try It 6.112 §6.4 source ↗

Find the product:

(a b - 9) (a b + 9) .

Answer

a^{2} b^{2} - 81

Try It 6.113 §6.4 source ↗

Find the product:

(3 x^{2} - 4 y^{3}) (3 x^{2} + 4 y^{3}) .

Answer

9 x^{4} - 16 y^{6}

Try It 6.114 §6.4 source ↗

Find the product:

(2 m^{2} - 5 n^{3}) (2 m^{2} + 5 n^{3}) .

Answer

4 m^{4} - 25 n^{6}

Try It 6.115 §6.4 source ↗

Choose the appropriate pattern and use it to find the product:

ⓐ $(9 b - 2) (2 b + 9)$ ⓑ ${(9 p - 4)}^{2}$ ⓒ ${(7 y + 1)}^{2}$ ⓓ $(4 r - 3) (4 r + 3)$

Answer

ⓐ FOIL;

18 b^{2} + 77 b - 18

ⓑ Binomial Squares;

81 p^{2} - 72 p + 16

49 y^{2} + 14 y + 1

ⓓ Product of Conjugates;

16 r^{2} - 9

Try It 6.116 §6.4 source ↗

Choose the appropriate pattern and use it to find the product:

ⓐ ${(6 x + 7)}^{2}$ ⓑ $(3 x - 4) (3 x + 4)$ ⓒ $(2 x - 5) (5 x - 2)$ ⓓ ${(6 n - 1)}^{2}$

Answer

ⓐ Binomial Squares;

36 x^{2} + 84 x + 49

ⓑ Product of Conjugates;

9 x^{2} - 16

10 x^{2} - 29 x + 10

ⓓ Binomial Squares;

36 n^{2} - 12 n + 1

6.4-303 §6.4 source ↗

Square a Binomial Using the Binomial Squares Pattern

In the following exercises, square each binomial using the Binomial Squares Pattern.

${(q + 12)}^{2}$

Answer

q^{2} + 24 q + 144

6.4-305 §6.4 source ↗

Square a Binomial Using the Binomial Squares Pattern

In the following exercises, square each binomial using the Binomial Squares Pattern.

${(x + \frac{2}{3})}^{2}$

Answer

x^{2} + \frac{4}{3} x + \frac{4}{9}

6.4-307 §6.4 source ↗

Square a Binomial Using the Binomial Squares Pattern

In the following exercises, square each binomial using the Binomial Squares Pattern.

${(y - 6)}^{2}$

Answer

y^{2} - 12 y + 36

6.4-309 §6.4 source ↗

Square a Binomial Using the Binomial Squares Pattern

In the following exercises, square each binomial using the Binomial Squares Pattern.

${(p - 13)}^{2}$

Answer

p^{2} - 26 p + 169

6.4-311 §6.4 source ↗

Square a Binomial Using the Binomial Squares Pattern

In the following exercises, square each binomial using the Binomial Squares Pattern.

${(4 a + 10)}^{2}$

Answer

16 a^{2} +

80 a +

100

6.4-313 §6.4 source ↗

Square a Binomial Using the Binomial Squares Pattern

In the following exercises, square each binomial using the Binomial Squares Pattern.

${(3 z + \frac{1}{5})}^{2}$

Answer

9 z^{2} + \frac{6}{5} z + \frac{1}{25}

6.4-315 §6.4 source ↗

Square a Binomial Using the Binomial Squares Pattern

In the following exercises, square each binomial using the Binomial Squares Pattern.

${(2 y - 3 z)}^{2}$

Answer

4 y^{2} -

12 y z +

9 z^{2}

6.4-317 §6.4 source ↗

Square a Binomial Using the Binomial Squares Pattern

In the following exercises, square each binomial using the Binomial Squares Pattern.

${(\frac{1}{8} x - \frac{1}{9} y)}^{2}$

Answer

\frac{1}{64} x^{2} -

\frac{1}{36} x y +

\frac{1}{81} y^{2}

6.4-319 §6.4 source ↗

Square a Binomial Using the Binomial Squares Pattern

In the following exercises, square each binomial using the Binomial Squares Pattern.

${(5 u^{2} + 9)}^{2}$

Answer

25 u^{4} +

90 u^{2} +

81

6.4-321 §6.4 source ↗

Square a Binomial Using the Binomial Squares Pattern

In the following exercises, square each binomial using the Binomial Squares Pattern.

${(8 p^{3} - 3)}^{2}$

Answer

64 p^{6} - 48 p^{3} + 9

6.4-323 §6.4 source ↗

Multiply Conjugates Using the Product of Conjugates Pattern

In the following exercises, multiply each pair of conjugates using the Product of Conjugates Pattern.

$(c - 5) (c + 5)$

Answer

c^{2} - 25

6.4-325 §6.4 source ↗

Multiply Conjugates Using the Product of Conjugates Pattern

In the following exercises, multiply each pair of conjugates using the Product of Conjugates Pattern.

$(b + \frac{6}{7}) (b - \frac{6}{7})$

Answer

b^{2} - \frac{36}{49}

6.4-327 §6.4 source ↗

Multiply Conjugates Using the Product of Conjugates Pattern

In the following exercises, multiply each pair of conjugates using the Product of Conjugates Pattern.

$(8 j + 4) (8 j - 4)$

Answer

64 j^{2} - 16

6.4-329 §6.4 source ↗

Multiply Conjugates Using the Product of Conjugates Pattern

In the following exercises, multiply each pair of conjugates using the Product of Conjugates Pattern.

$(9 c + 5) (9 c - 5)$

Answer

81 c^{2} - 25

6.4-331 §6.4 source ↗

Multiply Conjugates Using the Product of Conjugates Pattern

In the following exercises, multiply each pair of conjugates using the Product of Conjugates Pattern.

$(13 - q) (13 + q)$

Answer

169 - q^{2}

6.4-333 §6.4 source ↗

Multiply Conjugates Using the Product of Conjugates Pattern

In the following exercises, multiply each pair of conjugates using the Product of Conjugates Pattern.

$(4 - 6 y) (4 + 6 y)$

Answer

16 - 36 y^{2}

6.4-335 §6.4 source ↗

Multiply Conjugates Using the Product of Conjugates Pattern

In the following exercises, multiply each pair of conjugates using the Product of Conjugates Pattern.

$(7 w + 10 x) (7 w - 10 x)$

Answer

49 w^{2} - 100 x^{2}

6.4-337 §6.4 source ↗

Multiply Conjugates Using the Product of Conjugates Pattern

In the following exercises, multiply each pair of conjugates using the Product of Conjugates Pattern.

$(p + \frac{4}{5} q) (p - \frac{4}{5} q)$

Answer

p^{2} - \frac{16}{25} q^{2}

6.4-339 §6.4 source ↗

Multiply Conjugates Using the Product of Conjugates Pattern

In the following exercises, multiply each pair of conjugates using the Product of Conjugates Pattern.

$(x y - 9) (x y + 9)$

Answer

x^{2} y^{2} - 81

6.4-341 §6.4 source ↗

Multiply Conjugates Using the Product of Conjugates Pattern

In the following exercises, multiply each pair of conjugates using the Product of Conjugates Pattern.

$(r s - \frac{2}{7}) (r s + \frac{2}{7})$

Answer

r^{2} s^{2} - \frac{4}{49}

6.4-343 §6.4 source ↗

Multiply Conjugates Using the Product of Conjugates Pattern

In the following exercises, multiply each pair of conjugates using the Product of Conjugates Pattern.

$(6 m^{3} - 4 n^{5}) (6 m^{3} + 4 n^{5})$

Answer

36 m^{6} - 16 n^{10}

6.4-345 §6.4 source ↗

Multiply Conjugates Using the Product of Conjugates Pattern

In the following exercises, multiply each pair of conjugates using the Product of Conjugates Pattern.

$(15 m^{2} - 8 n^{4}) (15 m^{2} + 8 n^{4})$

Answer

225 m^{4} - 64 n^{8}

6.4-347 §6.4 source ↗

Recognize and Use the Appropriate Special Product Pattern

In the following exercises, find each product.

ⓐ ${(2 r + 12)}^{2}$ ⓑ $(3 p + 8) (3 p - 8)$ ⓒ $(7 a + b) (a - 7 b)$ ⓓ ${(k - 6)}^{2}$

Answer

ⓐ

4 r^{2} + 48 r + 144

ⓑ

9 p^{2} - 64

ⓒ

7 a^{2} -

48 a b -

7 b^{2}

ⓓ

k^{2} - 12 k + 36

6.4-349 §6.4 source ↗

Recognize and Use the Appropriate Special Product Pattern

In the following exercises, find each product.

ⓐ $(x^{5} + y^{5}) (x^{5} - y^{5})$ ⓑ ${(m^{3} - 8 n)}^{2}$ ⓒ ${(9 p + 8 q)}^{2}$ ⓓ $(r^{2} - s^{3}) (r^{3} + s^{2})$

Answer

ⓐ

x^{10} - y^{10}

ⓑ

m^{6} -

16 m^{3} n +

64 n^{2}

ⓒ

81 p^{2} +

144 p q +

64 q^{2}

ⓓ

r^{5} +

r^{2} s^{2} -

r^{3} s^{3} -

s^{5}

6.4-351 §6.4 source ↗

Recognize and Use the Appropriate Special Product Pattern

In the following exercises, find each product.

Mental math You can use the binomial squares pattern to multiply numbers without a calculator. Say you need to square 65. Think of 65 as $60 + 5$ .

ⓐ Multiply ${(60 + 5)}^{2}$ by using the binomial squares pattern, ${(a + b)}^{2} =$ $a^{2} +$ $2 a b +$ $b^{2}$ .

ⓑ Square 65 without using a calculator.

Answer

ⓐ 4,225 ⓑ 4,225 ⓒ Answers will vary.

6.4-353 §6.4 source ↗

Recognize and Use the Appropriate Special Product Pattern

In the following exercises, find each product.

Why does ${(a + b)}^{2}$ result in a trinomial, but $(a - b) (a + b)$ result in a binomial?

Answer

Answers will vary.

6.4-355 §6.4 source ↗

Recognize and Use the Appropriate Special Product Pattern

In the following exercises, find each product.

Use the order of operations to show that ${(3 + 5)}^{2}$ is 64, and then use that numerical example to explain why ${(a + b)}^{2} \neq a^{2} +$ $b^{2}$ .

Answer

Answers will vary.

Be Prepared 6.9 §6.5 source ↗

Simplify:

\frac{8}{24} .

If you missed this problem, review Example 1.65.

Example 1.65 in the book ↗

Answer

\frac{1}{3}

Be Prepared 6.10 §6.5 source ↗

Simplify:

{(2 m^{3})}^{5} .

If you missed this problem, review Example 6.23.

Example 6.23 in the book ↗

Answer

32 m^{15}

Be Prepared 6.11 §6.5 source ↗

Simplify:

\frac{12 x}{12 y} .

If you missed this problem, review Example 1.67.

Example 1.67 in the book ↗

Answer

\frac{x}{y}

Try It 6.117 §6.5 source ↗

Simplify: ⓐ

\frac{x^{15}}{x^{10}}

ⓑ

\frac{6^{14}}{6^{5}} .

Answer

ⓐ

x^{5}

ⓑ

6^{9}

Try It 6.118 §6.5 source ↗

Simplify: ⓐ

\frac{y^{43}}{y^{37}}

ⓑ

\frac{10^{15}}{10^{7}} .

Answer

ⓐ

y^{6}

ⓑ

10^{8}

Try It 6.119 §6.5 source ↗

Simplify: ⓐ

\frac{x^{18}}{x^{22}}

ⓑ

\frac{12^{15}}{12^{30}} .

Answer

ⓐ

\frac{1}{x^{4}}

ⓑ

\frac{1}{12^{15}}

Try It 6.120 §6.5 source ↗

Simplify: ⓐ

\frac{m^{7}}{m^{15}}

ⓑ

\frac{9^{8}}{9^{19}} .

Answer

ⓐ

\frac{1}{m^{8}}

ⓑ

\frac{1}{9^{11}}

Try It 6.121 §6.5 source ↗

Simplify: ⓐ

\frac{b^{19}}{b^{11}}

ⓑ

\frac{z^{5}}{z^{11}} .

Answer

ⓐ

b^{8}

ⓑ

\frac{1}{z^{6}}

Try It 6.122 §6.5 source ↗

Simplify: ⓐ

\frac{p^{9}}{p^{17}}

ⓑ

\frac{w^{13}}{w^{9}} .

Answer

ⓐ

\frac{1}{p^{8}}

ⓑ

w^{4}

Try It 6.123 §6.5 source ↗

Simplify: ⓐ

15^{0}

ⓑ

m^{0} .

Answer

ⓐ 1 ⓑ 1

Try It 6.124 §6.5 source ↗

Simplify: ⓐ

k^{0}

ⓑ

29^{0} .

Answer

ⓐ 1 ⓑ 1

Try It 6.125 §6.5 source ↗

Simplify: ⓐ

{(11 z)}^{0}

ⓑ

{(−11 p q^{3})}^{0} .

Answer

ⓐ

1

ⓑ

1

Try It 6.126 §6.5 source ↗

Simplify: ⓐ

{(−6 d)}^{0}

ⓑ

{(−8 m^{2} n^{3})}^{0} .

Answer

ⓐ

1

ⓑ

1

Try It 6.127 §6.5 source ↗

Simplify: ⓐ

{(\frac{5}{8})}^{2}

ⓑ

{(\frac{p}{10})}^{4}

ⓒ

{(\frac{m}{n})}^{7} .

Answer

ⓐ

\frac{25}{64}

ⓑ

\frac{p^{4}}{10,000}

ⓒ

\frac{m^{7}}{n^{7}}

Try It 6.128 §6.5 source ↗

Simplify: ⓐ

{(\frac{1}{3})}^{3}

ⓑ

{(\frac{−2}{q})}^{3}

ⓒ

{(\frac{w}{x})}^{4} .

Answer

ⓐ

\frac{1}{27}

ⓑ

\frac{−8}{q^{3}}

ⓒ

\frac{w^{4}}{x^{4}}

Try It 6.129 §6.5 source ↗

Simplify:

\frac{{(m^{5})}^{4}}{m^{7}} .

Answer

m^{13}

Try It 6.130 §6.5 source ↗

Simplify:

\frac{{(k^{2})}^{6}}{k^{7}} .

Answer

k^{5}

Try It 6.131 §6.5 source ↗

Simplify:

\frac{n^{12}}{{(n^{3})}^{4}} .

Answer

Try It 6.132 §6.5 source ↗

Simplify:

\frac{x^{15}}{{(x^{3})}^{5}} .

Answer

Try It 6.133 §6.5 source ↗

Simplify:

{(\frac{r^{5}}{r^{3}})}^{4} .

Answer

r^{8}

Try It 6.134 §6.5 source ↗

Simplify:

{(\frac{v^{6}}{v^{4}})}^{3} .

Answer

v^{6}

Try It 6.135 §6.5 source ↗

Simplify:

{(\frac{a^{3}}{b^{2}})}^{4} .

Answer

\frac{a^{12}}{b^{8}}

Try It 6.136 §6.5 source ↗

Simplify:

{(\frac{q^{7}}{r^{5}})}^{3} .

Answer

\frac{q^{21}}{r^{15}}

Try It 6.137 §6.5 source ↗

Simplify:

{(\frac{7 x^{3}}{9 y})}^{2} .

Answer

\frac{49 x^{6}}{81 y^{2}}

Try It 6.138 §6.5 source ↗

Simplify:

{(\frac{3 x^{4}}{7 y})}^{2} .

Answer

\frac{9 x^{8}}{49 y^{2}}

Try It 6.139 §6.5 source ↗

Simplify:

\frac{{(a^{2})}^{3} {(a^{2})}^{4}}{{(a^{4})}^{5}} .

Answer

\frac{1}{a^{6}}

Try It 6.140 §6.5 source ↗

Simplify:

\frac{{(p^{3})}^{4} {(p^{5})}^{3}}{{(p^{7})}^{6}} .

Answer

\frac{1}{p^{15}}

Try It 6.141 §6.5 source ↗

Simplify:

\frac{{(3 r^{3})}^{2} {(r^{3})}^{7}}{{(r^{3})}^{3}} .

Answer

9 r^{18}

Try It 6.142 §6.5 source ↗

Simplify:

\frac{{(2 x^{4})}^{5}}{{(4 x^{3})}^{2} {(x^{3})}^{5}} .

Answer

\frac{2}{x}

Try It 6.143 §6.5 source ↗

Find the quotient:

42 y^{9} \div 6 y^{3} .

Answer

7 y^{6}

Try It 6.144 §6.5 source ↗

Find the quotient:

48 z^{8} \div 8 z^{2} .

Answer

6 z^{6}

Try It 6.145 §6.5 source ↗

Find the quotient:

\frac{−72 a^{7} b^{3}}{8 a^{12} b^{4}} .

Answer

- \frac{9}{a^{5} b}

Try It 6.146 §6.5 source ↗

Find the quotient:

\frac{−63 c^{8} d^{3}}{7 c^{12} d^{2}} .

Answer

\frac{−9 d}{c^{4}}

Try It 6.147 §6.5 source ↗

Find the quotient:

\frac{16 a^{7} b^{6}}{24 a b^{8}} .

Answer

\frac{2 a^{6}}{3 b^{2}}

Try It 6.148 §6.5 source ↗

Find the quotient:

\frac{27 p^{4} q^{7}}{−45 p^{12} q} .

Answer

- \frac{3 q^{6}}{5 p^{8}}

Try It 6.149 §6.5 source ↗

Find the quotient:

\frac{28 x^{5} y^{14}}{49 x^{9} y^{12}} .

Answer

\frac{4 y^{2}}{7 x^{4}}

Try It 6.150 §6.5 source ↗

Find the quotient:

\frac{30 m^{5} n^{11}}{48 m^{10} n^{14}} .

Answer

\frac{5}{8 m^{5} n^{3}}

Try It 6.151 §6.5 source ↗

Find the quotient:

\frac{(6 a^{4} b^{5}) (4 a^{2} b^{5})}{12 a^{5} b^{8}} .

Answer

2 a b^{2}

Try It 6.152 §6.5 source ↗

Find the quotient:

\frac{(−12 x^{6} y^{9}) (−4 x^{5} y^{8})}{−12 x^{10} y^{12}} .

Answer

−4 x y^{5}

6.5-357 §6.5 source ↗

Simplify Expressions Using the Quotient Property for Exponents

In the following exercises, simplify.

ⓐ $\frac{y^{20}}{y^{10}}$ ⓑ $\frac{7^{16}}{7^{2}}$

Answer

ⓐ

y^{10}

ⓑ

7^{14}

6.5-359 §6.5 source ↗

Simplify Expressions Using the Quotient Property for Exponents

In the following exercises, simplify.

ⓐ $\frac{u^{24}}{u^{3}}$ ⓑ $\frac{9^{15}}{9^{5}}$

Answer

ⓐ

u^{21}

ⓑ

9^{10}

6.5-361 §6.5 source ↗

Simplify Expressions Using the Quotient Property for Exponents

In the following exercises, simplify.

ⓐ $\frac{t^{10}}{t^{40}}$ ⓑ $\frac{8^{3}}{8^{5}}$

Answer

ⓐ

\frac{1}{t^{30}}

ⓑ

\frac{1}{64}

6.5-363 §6.5 source ↗

Simplify Expressions Using the Quotient Property for Exponents

In the following exercises, simplify.

ⓐ $\frac{x}{x^{7}}$ ⓑ $\frac{10}{10^{3}}$

Answer

ⓐ

\frac{1}{x^{6}}

ⓑ

\frac{1}{100}

6.5-365 §6.5 source ↗

Simplify Expressions with Zero Exponents

In the following exercises, simplify.

ⓐ $13^{0}$ ⓑ $k^{0}$

Answer

ⓐ 1 ⓑ 1

6.5-367 §6.5 source ↗

Simplify Expressions with Zero Exponents

In the following exercises, simplify.

ⓐ $− 15^{0}$ ⓑ $− (15^{0})$

Answer

ⓐ

−1

ⓑ

−1

6.5-369 §6.5 source ↗

Simplify Expressions with Zero Exponents

In the following exercises, simplify.

ⓐ ${(6 y)}^{0}$ ⓑ $6 y^{0}$

Answer

ⓐ 1 ⓑ 6

6.5-371 §6.5 source ↗

Simplify Expressions with Zero Exponents

In the following exercises, simplify.

ⓐ $7 y^{0}$ ${(17 y)}^{0}$ ⓑ ${(−93 c^{7} d^{15})}^{0}$

Answer

ⓐ 7 ⓑ 1

6.5-373 §6.5 source ↗

Simplify Expressions with Zero Exponents

In the following exercises, simplify.

ⓐ $15 r^{0} - 22 s^{0}$ ⓑ ${(15 r)}^{0} -$ ${(22 s)}^{0}$

Answer

ⓐ

−7

ⓑ 0

6.5-375 §6.5 source ↗

Simplify Expressions Using the Quotient to a Power Property

In the following exercises, simplify.

ⓐ ${(\frac{2}{5})}^{2}$ ⓑ ${(\frac{x}{3})}^{4}$ ⓒ ${(\frac{a}{b})}^{5}$

Answer

ⓐ

\frac{4}{25}

ⓑ

\frac{x^{4}}{81}

ⓒ

\frac{a^{5}}{b^{5}}

6.5-377 §6.5 source ↗

Simplify Expressions Using the Quotient to a Power Property

In the following exercises, simplify.

ⓐ ${(\frac{x}{2 y})}^{3}$ ⓑ ${(\frac{10}{3 q})}^{4}$

Answer

ⓐ

\frac{x^{3}}{8 y^{3}}

ⓑ

\frac{10,000}{81 q^{4}}

6.5-379 §6.5 source ↗

Simplify Expressions by Applying Several Properties

In the following exercises, simplify.

$\frac{{(p^{3})}^{4}}{p^{5}}$

Answer

p^{7}

6.5-381 §6.5 source ↗

Simplify Expressions by Applying Several Properties

In the following exercises, simplify.

$\frac{{(x^{4})}^{5}}{x^{15}}$

Answer

x^{5}

6.5-383 §6.5 source ↗

Simplify Expressions by Applying Several Properties

In the following exercises, simplify.

$\frac{v^{20}}{{(v^{4})}^{5}}$

Answer

6.5-385 §6.5 source ↗

Simplify Expressions by Applying Several Properties

In the following exercises, simplify.

$\frac{n^{8}}{{(n^{6})}^{4}}$

Answer

\frac{1}{n^{16}}

6.5-387 §6.5 source ↗

Simplify Expressions by Applying Several Properties

In the following exercises, simplify.

${(\frac{q^{8}}{q^{2}})}^{3}$

Answer

q^{18}

6.5-389 §6.5 source ↗

Simplify Expressions by Applying Several Properties

In the following exercises, simplify.

${(\frac{m^{4}}{m^{7}})}^{4}$

Answer

\frac{1}{m^{12}}

6.5-391 §6.5 source ↗

Simplify Expressions by Applying Several Properties

In the following exercises, simplify.

${(\frac{a}{b^{6}})}^{3}$

Answer

\frac{a^{3}}{b^{18}}

6.5-393 §6.5 source ↗

Simplify Expressions by Applying Several Properties

In the following exercises, simplify.

${(\frac{y^{4}}{z^{10}})}^{5}$

Answer

\frac{y^{20}}{z^{50}}

6.5-395 §6.5 source ↗

Simplify Expressions by Applying Several Properties

In the following exercises, simplify.

${(\frac{3 m^{5}}{5 n})}^{3}$

Answer

\frac{27 m^{15}}{125 n^{3}}

6.5-397 §6.5 source ↗

Simplify Expressions by Applying Several Properties

In the following exercises, simplify.

${(\frac{5 u^{7}}{2 v^{3}})}^{4}$

Answer

\frac{625 u^{28}}{16 v^{^{12}}}

6.5-399 §6.5 source ↗

Simplify Expressions by Applying Several Properties

In the following exercises, simplify.

${(\frac{j^{2} j^{5}}{j^{4}})}^{3}$

Answer

j^{9}

6.5-401 §6.5 source ↗

Simplify Expressions by Applying Several Properties

In the following exercises, simplify.

$\frac{{(q^{3})}^{6} {(q^{2})}^{3}}{{(q^{4})}^{8}}$

Answer

\frac{1}{q^{8}}

6.5-403 §6.5 source ↗

Simplify Expressions by Applying Several Properties

In the following exercises, simplify.

$\frac{{(−2 k^{3})}^{2} {(6 k^{2})}^{4}}{{(9 k^{4})}^{2}}$

Answer

64 k^{6}

6.5-405 §6.5 source ↗

Simplify Expressions by Applying Several Properties

In the following exercises, simplify.

$\frac{{(−10 n^{2})}^{3} {(4 n^{5})}^{2}}{{(2 n^{8})}^{2}}$

Answer

−4,000

6.5-407 §6.5 source ↗

Divide Monomials

In the following exercises, divide the monomials.

$63 v^{10} \div 9 v^{2}$

Answer

7 v^{8}

6.5-409 §6.5 source ↗

Divide Monomials

In the following exercises, divide the monomials.

$−72 u^{12} \div 12 u^{4}$

Answer

−6 u^{8}

6.5-411 §6.5 source ↗

Divide Monomials

In the following exercises, divide the monomials.

$\frac{54 x^{9} y^{3}}{−18 x^{6} y^{15}}$

Answer

- \frac{3 x^{3}}{y^{12}}

6.5-413 §6.5 source ↗

Divide Monomials

In the following exercises, divide the monomials.

$\frac{20 m^{8} n^{4}}{30 m^{5} n^{9}}$

Answer

\frac{2 m^{3}}{3 n^{5}}

6.5-415 §6.5 source ↗

Divide Monomials

In the following exercises, divide the monomials.

$\frac{45 x^{5} y^{9}}{−60 x^{8} y^{6}}$

Answer

\frac{−3 y^{3}}{4 x^{3}}

6.5-417 §6.5 source ↗

Divide Monomials

In the following exercises, divide the monomials.

$\frac{65 a^{10} b^{8} c^{5}}{42 a^{7} b^{6} c^{8}}$

Answer

\frac{65 a^{3} b^{2}}{42 c^{3}}

6.5-419 §6.5 source ↗

Divide Monomials

In the following exercises, divide the monomials.

$\frac{(−18 p^{4} q^{7}) (−6 p^{3} q^{8})}{−36 p^{12} q^{10}}$

Answer

\frac{−3 q^{5}}{p^{5}}

6.5-421 §6.5 source ↗

Divide Monomials

In the following exercises, divide the monomials.

$\frac{(4 u^{2} v^{5}) (15 u^{3} v)}{(12 u^{3} v) (u^{4} v)}$

Answer

\frac{5 v^{4}}{u^{2}}

6.5-423 §6.5 source ↗

Mixed Practice

ⓐ $15 n^{10} + 3 n^{10}$ ⓑ $15 n^{10} - 3 n^{10}$ ⓒ $15 n^{10} \cdot 3 n^{10}$ ⓓ $15 n^{10} \div 3 n^{10}$

Answer

ⓐ

18 n^{10}

ⓑ

12 n^{10}

ⓒ

45 n^{20}

ⓓ 5

6.5-425 §6.5 source ↗

Mixed Practice

ⓐ $q^{5} \cdot q^{3}$ ⓑ ${(q^{5})}^{3}$

Answer

ⓐ

q^{8}

ⓑ

q^{15}

6.5-427 §6.5 source ↗

Mixed Practice

ⓐ $\frac{z^{6}}{z^{5}}$ ⓑ $\frac{z^{5}}{z^{6}}$

Answer

ⓐ

z

ⓑ

\frac{1}{z}

6.5-429 §6.5 source ↗

Mixed Practice

$(4 y) (12 y^{7}) \div 8 y^{2}$

Answer

6 y^{6}

6.5-431 §6.5 source ↗

Mixed Practice

$\frac{32 c^{11}}{4 c^{5}} +$ $\frac{42 c^{9}}{6 c^{3}}$

Answer

15 c^{6}

6.5-433 §6.5 source ↗

Mixed Practice

$\frac{48 x^{6}}{6 x^{4}} -$ $\frac{35 x^{9}}{7 x^{7}}$

Answer

3 x^{2}

6.5-435 §6.5 source ↗

Mixed Practice

$\frac{56 y^{4} z^{5}}{7 y^{3} z^{3}} -$ $\frac{45 y^{2} z^{2}}{5 y}$

Answer

− y z^{2}

6.5-437 §6.5 source ↗

Mixed Practice

Memory One gigabyte is approximately $10^{9}$ bytes. One terabyte is approximately $10^{12}$ bytes. How many gigabytes are in one terabyte?

Answer

10^{3}

6.5-439 §6.5 source ↗

Mixed Practice

Maurice simplifies the quotient $\frac{d^{7}}{d}$ by writing $\frac{d^{7}}{d} = 7$ . What is wrong with his reasoning?

Answer

Answers will vary.

6.5-441 §6.5 source ↗

Mixed Practice

Robert thinks $x^{0}$ simplifies to 0. What would you say to convince Robert he is wrong?

Answer

Answers will vary.

Be Prepared 6.12 §6.6 source ↗

Add:

\frac{3}{d} + \frac{x}{d} .

If you missed this problem, review Example 1.77.

Example 1.77 in the book ↗

Answer

\frac{3 + x}{d}

Be Prepared 6.13 §6.6 source ↗

Simplify:

\frac{30 x y^{3}}{5 x y} .

If you missed this problem, review Example 6.72.

Example 6.72 in the book ↗

Answer

6 y^{2}

Be Prepared 6.14 §6.6 source ↗

Combine like terms:

8 a^{2} +

12 a +

1 +

3 a^{2} -

5 a +

4 .

If you missed this problem, review Example 1.24.

Example 1.24 in the book ↗

Answer

11 a^{2} + 7 a + 5

Try It 6.153 §6.6 source ↗

Find the quotient:

\frac{8 z^{2} + 24}{4} .

Answer

2 z^{2} + 6

Try It 6.154 §6.6 source ↗

Find the quotient:

\frac{18 z^{2} - 27}{9} .

Answer

2 z^{2} - 3

Try It 6.155 §6.6 source ↗

Find the quotient:

(27 b^{3} - 33 b^{2}) \div 3 b .

Answer

9 b^{2} - 11 b

Try It 6.156 §6.6 source ↗

Find the quotient:

(25 y^{3} - 55 y^{2}) \div 5 y .

Answer

5 y^{2} - 11 y

Try It 6.157 §6.6 source ↗

Find the quotient:

\frac{25 y^{2} - 15 y}{−5} .

Answer

−5 y^{2} + 3 y

Try It 6.158 §6.6 source ↗

Find the quotient:

\frac{42 b^{2} - 18 b}{−6} .

Answer

−7 b^{2} + 3 b

Try It 6.159 §6.6 source ↗

Find the quotient:

\frac{60 d^{7} + 24 d^{5}}{4 d^{3}} .

Answer

15 d^{4} + 6 d^{2}

Try It 6.160 §6.6 source ↗

Find the quotient:

\frac{216 p^{7} - 48 p^{5}}{6 p^{3}} .

Answer

36 p^{4} - 8 p^{2}

Try It 6.161 §6.6 source ↗

Find the quotient:

(32 a^{2} b - 16 a b^{2}) \div (−8 a b) .

Answer

−4 a + 2 b

Try It 6.162 §6.6 source ↗

Find the quotient:

(−48 a^{8} b^{4} - 36 a^{6} b^{5}) \div (−6 a^{3} b^{3}) .

Answer

8 a^{5} b + 6 a^{3} b^{2}

Try It 6.163 §6.6 source ↗

Find the quotient:

\frac{40 x^{3} y^{2} + 24 x^{2} y^{2} - 16 x^{2} y^{3}}{8 x^{2} y} .

Answer

5 x y + 3 y - 2 y^{2}

Try It 6.164 §6.6 source ↗

Find the quotient:

\frac{35 a^{4} b^{2} + 14 a^{4} b^{3} - 42 a^{2} b^{4}}{7 a^{2} b^{2}} .

Answer

5 a^{2} +

2 a^{2} b -

6 b^{2}

Try It 6.165 §6.6 source ↗

Find the quotient:

\frac{18 c^{2} + 6 c - 9}{6 c} .

Answer

3 c + 1 - \frac{3}{2 c}

Try It 6.166 §6.6 source ↗

Find the quotient:

\frac{10 d^{2} - 5 d - 2}{5 d} .

Answer

2 d - 1 - \frac{2}{5 d}

Try It 6.167 §6.6 source ↗

Find the quotient:

(y^{2} + 10 y + 21) \div (y + 3) .

Answer

y + 7

Try It 6.168 §6.6 source ↗

Find the quotient:

(m^{2} + 9 m + 20) \div (m + 4) .

Answer

m + 5

Try It 6.169 §6.6 source ↗

Find the quotient:

(2 x^{2} - 3 x - 20) \div (x - 4) .

Answer

2 x + 5

Try It 6.170 §6.6 source ↗

Find the quotient:

(3 x^{2} - 16 x - 12) \div (x - 6) .

Answer

3 x + 2

Try It 6.171 §6.6 source ↗

Find the quotient:

(x^{3} + 5 x^{2} + 8 x + 6) \div (x + 2) .

Answer

x^{2} +

3 x +

2 +

\frac{2}{x + 2}

Try It 6.172 §6.6 source ↗

Find the quotient:

(2 x^{3} + 8 x^{2} + x - 8) \div (x + 1) .

Answer

2 x^{2} +

6 x -

5 -

\frac{3}{x + 1}

Try It 6.173 §6.6 source ↗

Find the quotient:

(x^{3} + 3 x + 14) \div (x + 2) .

Answer

x^{2} - 2 x + 7

Try It 6.174 §6.6 source ↗

Find the quotient:

(x^{4} - 3 x^{3} - 1000) \div (x + 5) .

Answer

x^{3} -

8 x^{2} +

40 x -

200

Try It 6.175 §6.6 source ↗

Find the quotient:

(x^{3} - 64) \div (x - 4) .

Answer

x^{2} + 4 x + 16

Try It 6.176 §6.6 source ↗

Find the quotient:

(125 x^{3} - 8) \div (5 x - 2) .

Answer

25 x^{2} + 10 x + 4

6.6-443 §6.6 source ↗

In the following exercises, divide each polynomial by the monomial.

$\frac{30 b + 75}{5}$

Answer

6 b + 15

6.6-445 §6.6 source ↗

In the following exercises, divide each polynomial by the monomial.

$\frac{42 x^{2} - 14 x}{7}$

Answer

6 x^{2} - 2 x

6.6-447 §6.6 source ↗

In the following exercises, divide each polynomial by the monomial.

$(55 w^{2} - 10 w) \div 5 w$

Answer

11 w - 2

6.6-449 §6.6 source ↗

In the following exercises, divide each polynomial by the monomial.

$(8 x^{3} + 6 x^{2}) \div 2 x$

Answer

4 x^{2} + 3 x

6.6-451 §6.6 source ↗

In the following exercises, divide each polynomial by the monomial.

$\frac{20 b^{2} - 12 b}{−4}$

Answer

−5 b^{2} + 3 b

6.6-453 §6.6 source ↗

In the following exercises, divide each polynomial by the monomial.

$\frac{51 m^{4} + 72 m^{3}}{−3}$

Answer

−17 m^{4} - 24 m^{3}

6.6-455 §6.6 source ↗

In the following exercises, divide each polynomial by the monomial.

$\frac{412 z^{8} - 48 z^{5}}{4 z^{3}}$

Answer

103 z^{5} - 12 z^{2}

6.6-457 §6.6 source ↗

In the following exercises, divide each polynomial by the monomial.

$\frac{51 y^{4} + 42 y^{2}}{3 y^{2}}$

Answer

17 y^{2} + 14

6.6-459 §6.6 source ↗

In the following exercises, divide each polynomial by the monomial.

$(35 x^{4} - 21 x) \div (−7 x)$

Answer

−5 x^{3} + 3

6.6-461 §6.6 source ↗

In the following exercises, divide each polynomial by the monomial.

$(48 y^{4} - 24 y^{3}) \div (−8 y^{2})$

Answer

−6 y^{2} + 3 y

6.6-463 §6.6 source ↗

In the following exercises, divide each polynomial by the monomial.

$(45 x^{3} y^{4} + 60 x y^{2}) \div (5 x y)$

Answer

9 x^{2} y^{3} + 12 y

6.6-465 §6.6 source ↗

In the following exercises, divide each polynomial by the monomial.

$\frac{49 c^{2} d^{2} - 70 c^{3} d^{3} - 35 c^{2} d^{4}}{7 c d^{2}}$

Answer

7 c -

10 c^{2} d -

5 c d^{2}

6.6-467 §6.6 source ↗

In the following exercises, divide each polynomial by the monomial.

$\frac{72 r^{5} s^{2} + 132 r^{4} s^{3} - 96 r^{3} s^{5}}{12 r^{2} s^{2}}$

Answer

6 r^{3} +

11 r^{2} s -

8 r s^{3}

6.6-469 §6.6 source ↗

In the following exercises, divide each polynomial by the monomial.

$\frac{12 q^{2} + 3 q - 1}{3 q}$

Answer

4 q + 1 - \frac{1}{3 q}

6.6-471 §6.6 source ↗

In the following exercises, divide each polynomial by the monomial.

$\frac{20 y^{2} + 12 y - 1}{−4 y}$

Answer

−5 y - 3 + \frac{1}{4 y}

6.6-473 §6.6 source ↗

In the following exercises, divide each polynomial by the monomial.

$\frac{63 a^{3} - 108 a^{2} + 99 a}{9 a^{2}}$

Answer

7 a - 12 + \frac{11}{a}

6.6-475 §6.6 source ↗

Divide a Polynomial by a Binomial

In the following exercises, divide each polynomial by the binomial.

$(d^{2} + 8 d + 12) \div (d + 2)$

Answer

d + 6

6.6-477 §6.6 source ↗

Divide a Polynomial by a Binomial

In the following exercises, divide each polynomial by the binomial.

$(a^{2} - 2 a - 35) \div (a + 5)$

Answer

a - 7

6.6-479 §6.6 source ↗

Divide a Polynomial by a Binomial

In the following exercises, divide each polynomial by the binomial.

$(x^{2} - 14 x + 49) \div (x - 7)$

Answer

x - 7

6.6-481 §6.6 source ↗

Divide a Polynomial by a Binomial

In the following exercises, divide each polynomial by the binomial.

$(4 x^{2} - 17 x - 15) \div (x - 5)$

Answer

4 x + 3

6.6-483 §6.6 source ↗

Divide a Polynomial by a Binomial

In the following exercises, divide each polynomial by the binomial.

$(p^{2} + 11 p + 16) \div (p + 8)$

Answer

p + 3 - \frac{8}{p + 8}

6.6-485 §6.6 source ↗

Divide a Polynomial by a Binomial

In the following exercises, divide each polynomial by the binomial.

$(x^{2} + 2 x - 30) \div (x - 5)$

Answer

x + 7 + \frac{5}{x - 5}

6.6-487 §6.6 source ↗

Divide a Polynomial by a Binomial

In the following exercises, divide each polynomial by the binomial.

$(2 n^{3} - 10 n + 24) \div (n + 3)$

Answer

2 n^{2} - 6 n + 8

6.6-489 §6.6 source ↗

Divide a Polynomial by a Binomial

In the following exercises, divide each polynomial by the binomial.

$(7 q^{3} - 5 q - 2) \div (q - 1)$

Answer

7 q^{2} + 7 q + 2

6.6-491 §6.6 source ↗

Divide a Polynomial by a Binomial

In the following exercises, divide each polynomial by the binomial.

$(m^{3} + 1000) \div (m + 10)$

Answer

m^{2} - 10 m + 100

6.6-493 §6.6 source ↗

Divide a Polynomial by a Binomial

In the following exercises, divide each polynomial by the binomial.

$(x^{3} - 216) \div (x - 6)$

Answer

x^{2} + 6 x + 36

6.6-495 §6.6 source ↗

Divide a Polynomial by a Binomial

In the following exercises, divide each polynomial by the binomial.

$(125 y^{3} - 64) \div (5 y - 4)$

Answer

25 y^{2} + 20 x + 16

6.6-497 §6.6 source ↗

Divide a Polynomial by a Binomial

In the following exercises, divide each polynomial by the binomial.

Handshakes At a company meeting, every employee shakes hands with every other employee. The number of handshakes is given by the expression $\frac{n^{2} - n}{2}$ , where $n$ represents the number of employees. How many handshakes will there be if there are 10 employees at the meeting?

Answer

6.6-499 §6.6 source ↗

Divide a Polynomial by a Binomial

In the following exercises, divide each polynomial by the binomial.

Divide $\frac{10 x^{2} + x - 12}{2 x}$ and explain with words how you get each term of the quotient.

Answer

Answers will vary.

Be Prepared 6.15 §6.7 source ↗

What is the place value of the

6

in the number

64,891

?
If you missed this problem, review Example 1.1.

Example 1.1 in the book ↗

Answer

Ten thousands

Be Prepared 6.16 §6.7 source ↗

Name the decimal:

0.0012 .

If you missed this problem, review Example 1.91.

Example 1.91 in the book ↗

Answer

Twelve ten thousandths

Be Prepared 6.17 §6.7 source ↗

Subtract:

5 - (−3) .

If you missed this problem, review Example 1.42.

Example 1.42 in the book ↗

Answer

8

Try It 6.177 §6.7 source ↗

Simplify: ⓐ

2^{−3}

ⓑ

10^{−7} .

Answer

ⓐ

\frac{1}{8}

ⓑ

\frac{1}{10^{7}}

Try It 6.178 §6.7 source ↗

Simplify: ⓐ

3^{−2}

ⓑ

10^{−4} .

Answer

ⓐ

\frac{1}{9}

ⓑ

\frac{1}{10,000}

Try It 6.179 §6.7 source ↗

Simplify: ⓐ

\frac{1}{p^{−8}}

ⓑ

\frac{1}{4^{−3}} .

Answer

ⓐ

p^{8}

ⓑ

64

Try It 6.180 §6.7 source ↗

Simplify: ⓐ

\frac{1}{q^{−7}}

ⓑ

\frac{1}{2^{−4}} .

Answer

ⓐ

q^{7}

ⓑ

16

Try It 6.181 §6.7 source ↗

Simplify: ⓐ

{(\frac{2}{3})}^{−4}

ⓑ

{(- \frac{6 m}{n})}^{−2} .

Answer

ⓐ

\frac{81}{16}

ⓑ

\frac{n^{2}}{36 m^{2}}

Try It 6.182 §6.7 source ↗

Simplify: ⓐ

{(\frac{3}{5})}^{−3}

ⓑ

{(- \frac{a}{2 b})}^{−4} .

Answer

ⓐ

\frac{125}{27}

ⓑ

\frac{16 b^{4}}{a^{4}}

Try It 6.183 §6.7 source ↗

Simplify: ⓐ

{(−5)}^{−2}

ⓑ

− 5^{−2}

ⓒ

{(- \frac{1}{5})}^{−2}

ⓓ

− {(\frac{1}{5})}^{−2} .

Answer

ⓐ

\frac{1}{25}

ⓑ

- \frac{1}{25}

−25

Try It 6.184 §6.7 source ↗

Simplify: ⓐ

{(−7)}^{−2}

ⓑ

− 7^{−2}

, ⓒ

{(- \frac{1}{7})}^{−2}

ⓓ

− {(\frac{1}{7})}^{−2} .

Answer

ⓐ

\frac{1}{49}

ⓑ

- \frac{1}{49}

−49

Try It 6.185 §6.7 source ↗

Simplify: ⓐ

6 \cdot 3^{−1}

ⓑ

{(6 \cdot 3)}^{−1} .

Answer

ⓐ

2

ⓑ

\frac{1}{18}

Try It 6.186 §6.7 source ↗

Simplify: ⓐ

8 \cdot 2^{−2}

ⓑ

{(8 \cdot 2)}^{−2} .

Answer

ⓐ 2 ⓑ

\frac{1}{256}

Try It 6.187 §6.7 source ↗

Simplify: ⓐ

y^{−7}

ⓑ

{(z^{3})}^{−5} .

Answer

ⓐ

\frac{1}{y^{7}}

ⓑ

\frac{1}{z^{15}}

Try It 6.188 §6.7 source ↗

Simplify: ⓐ

p^{−9}

ⓑ

{(q^{4})}^{−6} .

Answer

ⓐ

\frac{1}{p^{9}}

ⓑ

\frac{1}{q^{24}}

Try It 6.189 §6.7 source ↗

Simplify: ⓐ

8 p^{−1}

ⓑ

{(8 p)}^{−1}

ⓒ

{(−8 p)}^{−1} .

Answer

ⓐ

\frac{8}{p}

ⓑ

\frac{1}{8 p}

ⓒ

- \frac{1}{8 p}

Try It 6.190 §6.7 source ↗

Simplify: ⓐ

{11 q}^{−1}

ⓑ

{(11 q)}^{−1}

− {(11 q)}^{−1}

ⓒ

- \frac{1}{11 q}

Answer

ⓐ

\frac{1}{11 q}

ⓑ

\frac{1}{11 q}

- \frac{1}{11 q}

ⓒ

- \frac{11}{q}

Try It 6.191 §6.7 source ↗

Simplify: ⓐ

x^{−3} \cdot x^{7}

ⓑ

y^{−7} \cdot y^{2}

ⓒ

z^{−4} \cdot z^{−5} .

Answer

ⓐ

x^{4}

ⓑ

\frac{1}{y^{5}}

ⓒ

\frac{1}{z^{9}}

Try It 6.192 §6.7 source ↗

Simplify: ⓐ

a^{−1} \cdot a^{6}

ⓑ

b^{−8} \cdot b^{4}

ⓒ

c^{−8} \cdot c^{−7} .

Answer

ⓐ

a^{5}

ⓑ

\frac{1}{b^{4}}

ⓒ

\frac{1}{c^{15}}

Try It 6.193 §6.7 source ↗

Simplify:

(p^{6} q^{−2}) (p^{−9} q^{−1}) .

Answer

\frac{1}{p^{3} q^{3}}

Try It 6.194 §6.7 source ↗

Simplify:

(r^{5} s^{−3}) (r^{−7} s^{−5}) .

Answer

\frac{1}{r^{2} s^{8}}

Try It 6.195 §6.7 source ↗

Simplify:

(−6 c^{−6} d^{4}) (−5 c^{−2} d^{−1}) .

Answer

\frac{30 d^{3}}{c^{8}}

Try It 6.196 §6.7 source ↗

Simplify:

{(−4 x^{4})}^{−2} .

Answer

\frac{1}{16 x^{8}}

Try It 6.197 §6.7 source ↗

Simplify:

{(2 b^{3})}^{−4} .

Answer

\frac{1}{16 b^{12}}

Try It 6.198 §6.7 source ↗

Simplify:

{(8 a^{−4})}^{2} .

Answer

\frac{64}{a^{8}}

Try It 6.199 §6.7 source ↗

Simplify:

{(2 c^{−4})}^{3} .

Answer

\frac{8}{c^{12}}

Try It 6.200 §6.7 source ↗

Simplify:

\frac{x^{8}}{x^{−3}} .

Answer

x^{11}

Try It 6.201 §6.7 source ↗

Simplify:

\frac{y^{8}}{y^{−6}} .

Answer

y^{14}

Try It 6.202 §6.7 source ↗

Write in scientific notation:

96,000 .

Answer

9.6 \times 10^{4}

Try It 6.203 §6.7 source ↗

Write in scientific notation:

48,300 .

Answer

4.83 \times 10^{4}

Try It 6.204 §6.7 source ↗

Write in scientific notation:

0.0078 .

Answer

7.8 \times 10^{−3}

Try It 6.205 §6.7 source ↗

Write in scientific notation:

0.0129 .

Answer

1.29 \times 10^{−2}

Try It 6.206 §6.7 source ↗

Convert to decimal form:

1.3 \times 10^{3} .

Answer

1,300

Try It 6.207 §6.7 source ↗

Convert to decimal form:

9.25 \times 10^{4} .

Answer

92,500

Try It 6.208 §6.7 source ↗

Convert to decimal form:

1.2 \times 10^{−4} .

Answer

0.00012

Try It 6.209 §6.7 source ↗

Convert to decimal form:

7.5 \times 10^{−2} .

Answer

0.075

Try It 6.210 §6.7 source ↗

Multiply

(3 \times 10^{6}) (2 \times 10^{−8})

. Write answers in decimal form.

Answer

0.06

Try It 6.211 §6.7 source ↗

Multiply

(3 \times 10^{−2}) (3 \times 10^{−1})

. Write answers in decimal form.

Answer

0.009

Try It 6.212 §6.7 source ↗

Divide

\frac{8 \times 10^{4}}{2 \times 10^{−1}}

. Write answers in decimal form.

Answer

400,000

Try It 6.213 §6.7 source ↗

Divide

\frac{8 \times 10^{2}}{4 \times 10^{−2}}

. Write answers in decimal form.

Answer

20,000

6.7-501 §6.7 source ↗

Use the Definition of a Negative Exponent

In the following exercises, simplify.

ⓐ $3^{−4}$ ⓑ $10^{−2}$

Answer

ⓐ

\frac{1}{81}

ⓑ

\frac{1}{100}

6.7-503 §6.7 source ↗

Use the Definition of a Negative Exponent

In the following exercises, simplify.

ⓐ $2^{−8}$ ⓑ $10^{−2}$

Answer

ⓐ

\frac{1}{256}

ⓑ

\frac{1}{100}

6.7-505 §6.7 source ↗

Use the Definition of a Negative Exponent

In the following exercises, simplify.

ⓐ $\frac{1}{c^{−5}}$ ⓑ $\frac{1}{5^{−2}}$

Answer

ⓐ

c^{5}

ⓑ 25

6.7-507 §6.7 source ↗

Use the Definition of a Negative Exponent

In the following exercises, simplify.

ⓐ $\frac{1}{t^{−9}}$ ⓑ $\frac{1}{10^{−4}}$

Answer

ⓐ

t^{9}

ⓑ 10000

6.7-509 §6.7 source ↗

Use the Definition of a Negative Exponent

In the following exercises, simplify.

ⓐ ${(\frac{3}{10})}^{−2}$ ⓑ ${(- \frac{2}{c d})}^{−3}$

Answer

ⓐ

\frac{100}{9}

ⓑ

- \frac{c^{3} d^{3}}{8}

6.7-511 §6.7 source ↗

Use the Definition of a Negative Exponent

In the following exercises, simplify.

ⓐ ${(\frac{7}{2})}^{−3}$ ⓑ ${(- \frac{3}{x y^{2}})}^{−3}$

Answer

ⓐ

\frac{8}{343}

ⓑ

- \frac{x^{3} y^{6}}{27}

6.7-513 §6.7 source ↗

Use the Definition of a Negative Exponent

In the following exercises, simplify.

Answer

ⓐ

\frac{1}{49}

ⓑ

- \frac{1}{49}

−49

6.7-515 §6.7 source ↗

Use the Definition of a Negative Exponent

In the following exercises, simplify.

Answer

ⓐ

- \frac{1}{125}

ⓑ

−125

ⓒ

−125

ⓓ

- \frac{1}{125}

6.7-517 §6.7 source ↗

Use the Definition of a Negative Exponent

In the following exercises, simplify.

ⓐ $2 \cdot 5^{−1}$ ⓑ ${(2 \cdot 5)}^{−1}$

Answer

ⓐ

\frac{2}{5}

ⓑ

\frac{1}{10}

6.7-519 §6.7 source ↗

Use the Definition of a Negative Exponent

In the following exercises, simplify.

ⓐ $3 \cdot 4^{−2}$ ⓑ ${(3 \cdot 4)}^{−2}$

Answer

ⓐ

\frac{3}{16}

ⓑ

\frac{1}{144}

6.7-521 §6.7 source ↗

Use the Definition of a Negative Exponent

In the following exercises, simplify.

ⓐ $b^{−5}$ ⓑ ${(k^{2})}^{−5}$

Answer

ⓐ

\frac{1}{b^{5}}

ⓑ

\frac{1}{k^{10}}

6.7-523 §6.7 source ↗

Use the Definition of a Negative Exponent

In the following exercises, simplify.

ⓐ $s^{−8}$ ⓑ ${(a^{9})}^{−10}$

Answer

ⓐ

\frac{1}{s^{8}}

ⓑ

\frac{1}{a^{90}}

6.7-525 §6.7 source ↗

Use the Definition of a Negative Exponent

In the following exercises, simplify.

Answer

ⓐ

\frac{6}{r}

ⓑ

\frac{1}{6 r}

ⓒ

- \frac{1}{6 r}

6.7-527 §6.7 source ↗

Use the Definition of a Negative Exponent

In the following exercises, simplify.

Answer

ⓐ

\frac{1}{16 q^{4}}

ⓑ

\frac{2}{q^{4}}

ⓒ

- \frac{2}{q^{4}}

6.7-529 §6.7 source ↗

Simplify Expressions with Integer Exponents

In the following exercises, simplify.

Answer

ⓐ

\frac{1}{s^{4}}

ⓑ

\frac{1}{q^{5}}

ⓒ

\frac{1}{y^{7}}

6.7-531 §6.7 source ↗

Simplify Expressions with Integer Exponents

In the following exercises, simplify.

Answer

ⓐ 1 ⓑ

y^{6}

ⓒ

\frac{1}{y^{4}}

6.7-533 §6.7 source ↗

Simplify Expressions with Integer Exponents

In the following exercises, simplify.

$x^{4} \cdot x^{−2} \cdot x^{−3}$

Answer

\frac{1}{x}

6.7-535 §6.7 source ↗

Simplify Expressions with Integer Exponents

In the following exercises, simplify.

$(m^{3} n^{−3}) (m^{−5} n^{−1})$

Answer

\frac{1}{m^{2} n^{4}}

6.7-537 §6.7 source ↗

Simplify Expressions with Integer Exponents

In the following exercises, simplify.

$(p q^{−4}) (p^{−6} q^{−3})$

Answer

\frac{1}{p^{5} q^{7}}

6.7-539 §6.7 source ↗

Simplify Expressions with Integer Exponents

In the following exercises, simplify.

$(−2 j^{−5} k^{8}) (7 j^{2} k^{−3})$

Answer

- \frac{14 k^{5}}{j^{3}}

6.7-541 §6.7 source ↗

Simplify Expressions with Integer Exponents

In the following exercises, simplify.

$(−5 m^{4} n^{6}) (8 m^{−5} n^{−3})$

Answer

- \frac{40 n^{3}}{m}

6.7-543 §6.7 source ↗

Simplify Expressions with Integer Exponents

In the following exercises, simplify.

${(4 y^{3})}^{−3}$

Answer

\frac{1}{64 y^{9}}

6.7-545 §6.7 source ↗

Simplify Expressions with Integer Exponents

In the following exercises, simplify.

${(2 p^{−5})}^{2}$

Answer

\frac{4}{p^{10}}

6.7-547 §6.7 source ↗

Simplify Expressions with Integer Exponents

In the following exercises, simplify.

$\frac{n^{5}}{n^{−2}}$

Answer

n^{7}

6.7-549 §6.7 source ↗

Simplify Expressions with Integer Exponents

In the following exercises, simplify.

$\frac{y^{−5}}{y^{−10}}$

Answer

y^{5}

6.7-551 §6.7 source ↗

Convert from Decimal Notation to Scientific Notation

In the following exercises, write each number in scientific notation.

340,000

Answer

3.4 \times 10^{5}

6.7-553 §6.7 source ↗

Convert from Decimal Notation to Scientific Notation

In the following exercises, write each number in scientific notation.

1,290,000

Answer

1.29 \times 10^{6}

6.7-555 §6.7 source ↗

Convert from Decimal Notation to Scientific Notation

In the following exercises, write each number in scientific notation.

0.041

Answer

4.1 \times 10^{−2}

6.7-557 §6.7 source ↗

Convert from Decimal Notation to Scientific Notation

In the following exercises, write each number in scientific notation.

0.00000103

Answer

1.03 \times 10^{−6}

6.7-559 §6.7 source ↗

Convert Scientific Notation to Decimal Form

In the following exercises, convert each number to decimal form.

$8.3 \times 10^{2}$

Answer

830

6.7-561 §6.7 source ↗

Convert Scientific Notation to Decimal Form

In the following exercises, convert each number to decimal form.

$1.6 \times 10^{10}$

Answer

16,000,000,000

6.7-563 §6.7 source ↗

Convert Scientific Notation to Decimal Form

In the following exercises, convert each number to decimal form.

$3.8 \times 10^{−2}$

Answer

0.038

6.7-565 §6.7 source ↗

Convert Scientific Notation to Decimal Form

In the following exercises, convert each number to decimal form.

$1.93 \times 10^{−5}$

Answer

0.0000193

6.7-567 §6.7 source ↗

Multiply and Divide Using Scientific Notation

In the following exercises, multiply. Write your answer in decimal form.

$(2 \times 10^{2}) (1 \times 10^{−4})$

Answer

0.02

6.7-569 §6.7 source ↗

Multiply and Divide Using Scientific Notation

In the following exercises, multiply. Write your answer in decimal form.

$(3.5 \times 10^{−4}) (1.6 \times 10^{−2})$

Answer

0.0000056

6.7-571 §6.7 source ↗

In the following exercises, divide. Write your answer in decimal form.

$\frac{5 \times 10^{−2}}{1 \times 10^{−10}}$

Answer

500,000,000

6.7-573 §6.7 source ↗

In the following exercises, divide. Write your answer in decimal form.

$\frac{8 \times 10^{6}}{4 \times 10^{−1}}$

Answer

20,000,000

6.7-575 §6.7 source ↗

In the following exercises, divide. Write your answer in decimal form.

The population of the world on July 4, 2010 was more than 6,850,000,000. Write the number in scientific notation

Answer

6.85 \times 10^{9}

6.7-577 §6.7 source ↗

In the following exercises, divide. Write your answer in decimal form.

The probability of winning the 2010 Megamillions lottery was about 0.0000000057. Write the number in scientific notation.

Answer

5.7 \times 10^{−9}

6.7-579 §6.7 source ↗

In the following exercises, divide. Write your answer in decimal form.

At the start of 2012, the US federal budget had a deficit of more than $$ 1.5 \times 10^{13}$ . Convert this number to decimal form.

Answer

15,000,000,000,000

6.7-581 §6.7 source ↗

In the following exercises, divide. Write your answer in decimal form.

The width of a proton is $1 \times 10^{−5}$ of the width of an atom. Convert this number to decimal form.

Answer

0.00001

6.7-583 §6.7 source ↗

In the following exercises, divide. Write your answer in decimal form.

Coin production In 1942, the U.S. Mint produced 154,500,000 nickels. Write 154,500,000 in scientific notation.

Answer

1.545 \times 10^{8}

6.7-585 §6.7 source ↗

In the following exercises, divide. Write your answer in decimal form.

Debt At the end of fiscal year 2015 the gross United States federal government debt was estimated to be approximately $18,600,000,000,000 ($18.6 trillion), according to the Federal Budget. The population of the United States was approximately 300,000,000 people at the end of fiscal year 2015.

ⓐ Write the debt in scientific notation.

ⓑ Write the population in scientific notation.

Answer

ⓐ

1.86 \times 10^{13}

ⓑ

3 \times 10^{8}

ⓒ

6.2 \times 10^{4}

6.7-587 §6.7 source ↗

In the following exercises, divide. Write your answer in decimal form.

When you convert a number from decimal notation to scientific notation, how do you know if the exponent will be positive or negative?

Answer

answers will vary

6.R-589 §6.R source ↗

Identify Polynomials, Monomials, Binomials and Trinomials

In the following exercises, determine if each of the following polynomials is a monomial, binomial, trinomial, or other polynomial.

Answer

6.R-591 §6.R source ↗

Determine the Degree of Polynomials

In the following exercises, determine the degree of each polynomial.

ⓐ $5 p^{3} -$ $8 p^{2} +$ $10 p -$ $4$

ⓑ $−20 q^{4}$

ⓓ $23 r^{2} s^{2} -$ $4 r s +$ $5$

ⓔ 100

Answer

6.R-593 §6.R source ↗

Add and Subtract Monomials

In the following exercises, add or subtract the monomials.

$−14 k + 19 k$

Answer

5 k

6.R-595 §6.R source ↗

Add and Subtract Monomials

In the following exercises, add or subtract the monomials.

$−9 c - 18 c$

Answer

−27 c

6.R-597 §6.R source ↗

Add and Subtract Monomials

In the following exercises, add or subtract the monomials.

$3 m^{2} + 7 n^{2} - 3 m^{2}$

Answer

7 n^{2}

6.R-599 §6.R source ↗

Add and Subtract Monomials

In the following exercises, add or subtract the monomials.

$13a + b$

Answer

13a + b

6.R-601 §6.R source ↗

Add and Subtract Polynomials

In the following exercises, add or subtract the polynomials.

$(9 p^{2} - 5 p + 3) +$ $(4 p^{2} - 4)$

Answer

13 p^{2} - 5 p - 1

6.R-603 §6.R source ↗

Add and Subtract Polynomials

In the following exercises, add or subtract the polynomials.

$(7 y^{2} - 8 y) -$ $(y - 4)$

Answer

7 y^{2} - 9 y + 4

6.R-605 §6.R source ↗

Add and Subtract Polynomials

In the following exercises, add or subtract the polynomials.

Find the sum of $(a^{2} + 6 a + 9) and (5 a^{3} - 7)$

Answer

5 a^{3} + a^{2} + 6 a + 2

6.R-607 §6.R source ↗

Evaluate a Polynomial for a Given Value of the Variable

In the following exercises, evaluate each polynomial for the given value.

Evaluate $10 - 12 x$ when:

ⓐ $x = 3$

ⓑ $x = 0$

Answer

ⓐ

−26

ⓑ 10 ⓒ 22

6.R-609 §6.R source ↗

Evaluate a Polynomial for a Given Value of the Variable

In the following exercises, evaluate each polynomial for the given value.

A manufacturer of stereo sound speakers has found that the revenue received from selling the speakers at a cost of p dollars each is given by the polynomial $−4 p^{2} + 460 p .$ Find the revenue received when $p = 75$ dollars.

Answer

12,000

6.R-611 §6.R source ↗

Simplify Expressions with Exponents

In the following exercises, simplify.

$17^{1}$

Answer

6.R-613 §6.R source ↗

Simplify Expressions with Exponents

In the following exercises, simplify.

${(0.5)}^{3}$

Answer

0.125

6.R-615 §6.R source ↗

Simplify Expressions with Exponents

In the following exercises, simplify.

$− 2^{6}$

Answer

−64

6.R-617 §6.R source ↗

Simplify Expressions Using the Product Property for Exponents

In the following exercises, simplify each expression.

$p^{15} \cdot p^{16}$

Answer

p^{31}

6.R-619 §6.R source ↗

Simplify Expressions Using the Product Property for Exponents

In the following exercises, simplify each expression.

$8 \cdot 8^{5}$

Answer

8^{6}

6.R-621 §6.R source ↗

Simplify Expressions Using the Product Property for Exponents

In the following exercises, simplify each expression.

$y^{c} \cdot y^{3}$

Answer

y^{c + 3}

6.R-623 §6.R source ↗

Simplify Expressions Using the Power Property for Exponents

In the following exercises, simplify each expression.

${(5^{3})}^{2}$

Answer

5^{6}

6.R-625 §6.R source ↗

Simplify Expressions Using the Power Property for Exponents

In the following exercises, simplify each expression.

${(3^{r})}^{s}$

Answer

3^{r s}

6.R-627 §6.R source ↗

Simplify Expressions Using the Product to a Power Property

In the following exercises, simplify each expression.

${(−5 y)}^{3}$

Answer

−125 y^{3}

6.R-629 §6.R source ↗

Simplify Expressions Using the Product to a Power Property

In the following exercises, simplify each expression.

${(10 x y z)}^{3}$

Answer

1000 x^{3} y^{3} z^{3}

6.R-631 §6.R source ↗

Simplify Expressions by Applying Several Properties

In the following exercises, simplify each expression.

${(4 a^{3} b^{2})}^{3}$

Answer

64 a^{9} b^{6}

6.R-633 §6.R source ↗

Simplify Expressions by Applying Several Properties

In the following exercises, simplify each expression.

${(2 q^{3})}^{4} {(3 q)}^{2}$

Answer

144 q^{14}

6.R-635 §6.R source ↗

Simplify Expressions by Applying Several Properties

In the following exercises, simplify each expression.

${(\frac{2}{5} m^{2} n)}^{3}$

Answer

\frac{8}{125} m^{6} n^{3}

6.R-637 §6.R source ↗

Multiply Monomials

In the following exercises 8, multiply the monomials.

$(−9 n^{7}) (−16 n)$

Answer

144 n^{8}

6.R-639 §6.R source ↗

Multiply Monomials

In the following exercises 8, multiply the monomials.

$(\frac{5}{9} a b^{2}) (27 a b^{3})$

Answer

15 a^{2} b^{5}

6.R-641 §6.R source ↗

Multiply a Polynomial by a Monomial

In the following exercises, multiply.

$−4 (y + 13)$

Answer

−4 y - 52

6.R-643 §6.R source ↗

Multiply a Polynomial by a Monomial

In the following exercises, multiply.

$p (p + 3)$

Answer

p^{2} + 3 p

6.R-645 §6.R source ↗

Multiply a Polynomial by a Monomial

In the following exercises, multiply.

$−6 u (2 u + 7)$

Answer

−12 u^{2} - 42 u

6.R-647 §6.R source ↗

Multiply a Polynomial by a Monomial

In the following exercises, multiply.

$3 q^{2} (q^{2} - 7 q + 6)$ 3

Answer

9 q^{4} -

63 q^{3} +

54 q^{2}

6.R-649 §6.R source ↗

Multiply a Polynomial by a Monomial

In the following exercises, multiply.

$(b - 4) \cdot 11$

Answer

11 b - 44

6.R-651 §6.R source ↗

Multiply a Binomial by a Binomial

In the following exercises, multiply the binomials using: ⓐ the Distributive Property, ⓑ the FOIL method, ⓒ the Vertical Method.

$(6 y - 7) (2 y - 5)$

Answer

ⓐ

12 y^{2} - 44 y + 35

ⓑ

12 y^{2} - 44 y + 35

ⓒ

12 y^{2} - 44 y + 35

6.R-653 §6.R source ↗

In the following exercises, multiply the binomials. Use any method.

$(y - 4) (y - 8)$

Answer

y^{2} - 12 y + 32

6.R-655 §6.R source ↗

In the following exercises, multiply the binomials. Use any method.

$(q + 16) (q - 3)$

Answer

q^{2} + 13 q - 48

6.R-657 §6.R source ↗

In the following exercises, multiply the binomials. Use any method.

$(u^{2} + 6) (u^{2} - 5)$

Answer

u^{4} + u^{2} - 30

6.R-659 §6.R source ↗

In the following exercises, multiply the binomials. Use any method.

$(8 m n + 3) (2 m n - 1)$

Answer

16 m^{2} n^{2} -

2 m n -

3

6.R-661 §6.R source ↗

Multiply a Trinomial by a Binomial

In the following exercises, multiply using ⓐ the Distributive Property, ⓑ the Vertical Method.

$(3 x - 4) (6 x^{2} + x - 10)$

Answer

ⓐ

18 x^{3} -

21 x^{2} -

34 x +

40

ⓑ

18 x^{3} -

21 x^{2} -

34 x +

40

6.R-663 §6.R source ↗

In the following exercises, multiply. Use either method.

$(7 m + 1) (m^{2} - 10 m - 3)$

Answer

7 m^{3} -

69 m^{2} -

31 m -

3

6.R-665 §6.R source ↗

Square a Binomial Using the Binomial Squares Pattern

In the following exercises, square each binomial using the Binomial Squares Pattern.

${(q - 15)}^{2}$

Answer

q^{2} - 30 q + 225

6.R-667 §6.R source ↗

Square a Binomial Using the Binomial Squares Pattern

In the following exercises, square each binomial using the Binomial Squares Pattern.

${(8 u + 1)}^{2}$

Answer

64 u^{2} + 16 u + 1

6.R-669 §6.R source ↗

Square a Binomial Using the Binomial Squares Pattern

In the following exercises, square each binomial using the Binomial Squares Pattern.

${(4 a - 3 b)}^{2}$

Answer

16 a^{2} -

24 a b +

9 b^{2}

6.R-671 §6.R source ↗

Multiply Conjugates Using the Product of Conjugates Pattern

In the following exercises, multiply each pair of conjugates using the Product of Conjugates Pattern.

$(y + \frac{2}{5}) (y - \frac{2}{5})$

Answer

y^{2} - \frac{4}{25}

6.R-673 §6.R source ↗

Multiply Conjugates Using the Product of Conjugates Pattern

In the following exercises, multiply each pair of conjugates using the Product of Conjugates Pattern.

$(6 - r) (6 + r)$

Answer

36 - r^{2}

6.R-675 §6.R source ↗

Multiply Conjugates Using the Product of Conjugates Pattern

In the following exercises, multiply each pair of conjugates using the Product of Conjugates Pattern.

$(5 p^{4} - 4 q^{3}) (5 p^{4} + 4 q^{3})$

Answer

25 p^{8} - 16 q^{6}

6.R-677 §6.R source ↗

Recognize and Use the Appropriate Special Product Pattern

In the following exercises, find each product.

$(6 a + 11) (6 a - 11)$

Answer

36 a^{2} - 121

6.R-679 §6.R source ↗

Recognize and Use the Appropriate Special Product Pattern

In the following exercises, find each product.

${(c^{4} + 9 d)}^{2}$

Answer

c^{8} +

18 c^{4} d +

81 d^{2}

6.R-681 §6.R source ↗

Recognize and Use the Appropriate Special Product Pattern

In the following exercises, find each product.

$(a^{2} + 4 b) (4 a - b^{2})$

Answer

4 a^{3} -

a^{2} b^{2} +

16 a b -

4 b^{3}

6.R-683 §6.R source ↗

Simplify Expressions Using the Quotient Property for Exponents

In the following exercises, simplify.

$\frac{10^{25}}{10^{5}}$

Answer

10^{20}

6.R-685 §6.R source ↗

Simplify Expressions Using the Quotient Property for Exponents

In the following exercises, simplify.

$\frac{v^{12}}{v^{48}}$

Answer

\frac{1}{v^{36}}

6.R-687 §6.R source ↗

Simplify Expressions Using the Quotient Property for Exponents

In the following exercises, simplify.

$\frac{5}{5^{8}}$

Answer

\frac{1}{5^{7}}

6.R-689 §6.R source ↗

Simplify Expressions with Zero Exponents

In the following exercises, simplify.

$x^{0}$

Answer

6.R-691 §6.R source ↗

Simplify Expressions with Zero Exponents

In the following exercises, simplify.

$(− 12^{0})$ ${(−12)}^{0}$

Answer

−1

6.R-693 §6.R source ↗

Simplify Expressions with Zero Exponents

In the following exercises, simplify.

${(25 x)}^{0}$

Answer

6.R-695 §6.R source ↗

Simplify Expressions with Zero Exponents

In the following exercises, simplify.

${(19 n)}^{0} -$ ${(25 m)}^{0}$

Answer

6.R-697 §6.R source ↗

Simplify Expressions Using the Quotient to a Power Property

In the following exercises, simplify.

${(\frac{m}{3})}^{4}$

Answer

\frac{m^{4}}{81}

6.R-699 §6.R source ↗

Simplify Expressions Using the Quotient to a Power Property

In the following exercises, simplify.

${(\frac{x}{2 y})}^{6}$

Answer

\frac{x^{6}}{64 y^{6}}

6.R-701 §6.R source ↗

Simplify Expressions by Applying Several Properties

In the following exercises, simplify.

$\frac{n^{10}}{{(n^{5})}^{2}}$

Answer

6.R-703 §6.R source ↗

Simplify Expressions by Applying Several Properties

In the following exercises, simplify.

${(\frac{r^{8}}{r^{3}})}^{4}$

Answer

r^{20}

6.R-705 §6.R source ↗

Simplify Expressions by Applying Several Properties

In the following exercises, simplify.

${(\frac{3 x^{4}}{2 y^{2}})}^{5}$

Answer

\frac{243 x^{20}}{32 y^{10}}

6.R-707 §6.R source ↗

Simplify Expressions by Applying Several Properties

In the following exercises, simplify.

$\frac{{(3 n^{2})}^{4} {(−5 n^{4})}^{3}}{{(−2 n^{5})}^{2}}$

Answer

- \frac{10,125 n^{10}}{4}

6.R-709 §6.R source ↗

Divide Monomials

In the following exercises, divide the monomials.

$\frac{64 a^{5} b^{9}}{−16 a^{10} b^{3}}$

Answer

- \frac{4 b^{6}}{a^{5}}

6.R-711 §6.R source ↗

Divide Monomials

In the following exercises, divide the monomials.

$\frac{(8 p^{6} q^{2}) (9 p^{3} q^{5})}{16 p^{8} q^{7}}$

Answer

\frac{9 p}{2}

6.R-713 §6.R source ↗

Divide a Polynomial by a Monomial

In the following exercises, divide each polynomial by the monomial.

$(35 x^{2} - 75 x) \div 5 x$

Answer

7 x - 15

6.R-715 §6.R source ↗

Divide a Polynomial by a Monomial

In the following exercises, divide each polynomial by the monomial.

$\frac{550 p^{6} - 300 p^{4}}{10 p^{3}}$

Answer

55 p^{3} - 30 p

6.R-717 §6.R source ↗

Divide a Polynomial by a Monomial

In the following exercises, divide each polynomial by the monomial.

$\frac{96 a^{5} b^{2} - 48 a^{4} b^{3} - 56 a^{2} b^{4}}{8 a b^{2}}$

Answer

12 a^{4} -

6 a^{3} b -

7 a b^{2}

6.R-719 §6.R source ↗

Divide a Polynomial by a Monomial

In the following exercises, divide each polynomial by the monomial.

$\frac{105 y^{5} + 50 y^{3} - 5 y}{5 y^{3}}$

Answer

21 y^{2} + 10 - \frac{1}{y^{2}}

6.R-721 §6.R source ↗

Divide a Polynomial by a Binomial

In the following exercises, divide each polynomial by the binomial.

$(v^{2} - 16 v + 64) \div (v - 8)$

Answer

v - 8

6.R-723 §6.R source ↗

Divide a Polynomial by a Binomial

In the following exercises, divide each polynomial by the binomial.

$(n^{2} - 3 n - 14) \div (n + 3)$

Answer

n - 6 + \frac{4}{n + 3}

6.R-725 §6.R source ↗

Divide a Polynomial by a Binomial

In the following exercises, divide each polynomial by the binomial.

$(u^{3} - 8) \div (u - 2)$

Answer

u^{2} + 2 u + 4

6.R-727 §6.R source ↗

Use the Definition of a Negative Exponent

In the following exercises, simplify.

${(−5)}^{−3}$

Answer

- \frac{1}{125}

6.R-729 §6.R source ↗

Use the Definition of a Negative Exponent

In the following exercises, simplify.

${(6 u)}^{−3}$

Answer

\frac{1}{216 u^{3}}

6.R-731 §6.R source ↗

Use the Definition of a Negative Exponent

In the following exercises, simplify.

${(\frac{3}{4})}^{−2}$

Answer

\frac{16}{9}

6.R-733 §6.R source ↗

Simplify Expressions with Integer Exponents

In the following exercises, simplify.

$q^{−6} \cdot q^{−5}$

Answer

\frac{1}{q^{11}}

6.R-735 §6.R source ↗

Simplify Expressions with Integer Exponents

In the following exercises, simplify.

${(y^{8})}^{−1}$

Answer

\frac{1}{y^{8}}

6.R-737 §6.R source ↗

Simplify Expressions with Integer Exponents

In the following exercises, simplify.

$\frac{a^{8}}{a^{12}}$

Answer

\frac{1}{a^{4}}

6.R-739 §6.R source ↗

Simplify Expressions with Integer Exponents

In the following exercises, simplify.

$\frac{r^{−2}}{r^{−3}}$

Answer

r

6.R-741 §6.R source ↗

Convert from Decimal Notation to Scientific Notation

In the following exercises, write each number in scientific notation.

0.00429

Answer

4.29 \times 10^{−3}

6.R-743 §6.R source ↗

Convert from Decimal Notation to Scientific Notation

In the following exercises, write each number in scientific notation.

In 2015, the population of the world was about 7,200,000,000 people.

Answer

7.2 \times 10^{9}

6.R-745 §6.R source ↗

Convert Scientific Notation to Decimal Form

In the following exercises, convert each number to decimal form.

$1.5 \times 10^{10}$

Answer

15,000,000,000

6.R-747 §6.R source ↗

Convert Scientific Notation to Decimal Form

In the following exercises, convert each number to decimal form.

$5.5 \times 10^{−1}$

Answer

0.55

6.R-749 §6.R source ↗

Multiply and Divide Using Scientific Notation

In the following exercises, multiply and write your answer in decimal form.

$(3.5 \times 10^{−2}) (6.2 \times 10^{−1})$

Answer

0.0217

6.R-751 §6.R source ↗

In the following exercises, divide and write your answer in decimal form.

$\frac{9 \times 10^{−5}}{3 \times 10^{2}}$

Answer

0.0000003

6.T-753 §6.T source ↗

In the following exercises, simplify each expression.

$(12 a^{2} - 7 a + 4) +$ $(3 a^{2} + 8 a - 10)$

Answer

15 a^{2} + a - 6

6.T-755 §6.T source ↗

In the following exercises, simplify each expression.

${(- \frac{2}{5})}^{3}$

Answer

- \frac{8}{125}

6.T-757 §6.T source ↗

In the following exercises, simplify each expression.

${(4 a^{3} b^{5})}^{2}$

Answer

16 a^{6} b^{10}

6.T-759 §6.T source ↗

In the following exercises, simplify each expression.

$3 k (k^{2} - 7 k + 13)$

Answer

3 k^{3} -

21 k^{2} +

39 k

6.T-761 §6.T source ↗

In the following exercises, simplify each expression.

$(v - 9) (9 v - 5)$

Answer

9 v^{2} - 86 v + 45

6.T-763 §6.T source ↗

In the following exercises, simplify each expression.

$(n - 6) (n^{2} - 5 n + 4)$

Answer

n^{3} -

11 n^{2} +

34 n -

24

6.T-765 §6.T source ↗

In the following exercises, simplify each expression.

$(7 p - 5) (7 p + 5)$

Answer

49 p^{2} - 25

6.T-767 §6.T source ↗

In the following exercises, simplify each expression.

$\frac{3^{8}}{3^{10}}$

Answer

\frac{1}{9}

6.T-769 §6.T source ↗

In the following exercises, simplify each expression.

${(87 x^{15} y^{3} z^{22})}^{0}$

Answer

1

6.T-771 §6.T source ↗

In the following exercises, simplify each expression.

$\frac{12 x^{2} + 42 x - 6}{2 x}$

Answer

6 x + 21 - \frac{3}{x}

6.T-773 §6.T source ↗

In the following exercises, simplify each expression.

$\frac{64 x^{3} - 1}{4 x - 1}$

Answer

16 x^{2} + 4 x + 1

6.T-775 §6.T source ↗

In the following exercises, simplify each expression.

$5^{−2}$

Answer

\frac{1}{25}

6.T-777 §6.T source ↗

In the following exercises, simplify each expression.

$q^{−4} \cdot q^{−5}$

Answer

\frac{1}{q^{9}}

6.T-779 §6.T source ↗

In the following exercises, simplify each expression.

Convert 83,000,000 to scientific notation.

Answer

8.3 \times 10^{7}

6.T-781 §6.T source ↗

In the following exercises, simplify, and write your answer in decimal form.

$(3.4 \times 10^{9}) (2.2 \times 10^{−5})$

Answer

74,800

6.T-783 §6.T source ↗

In the following exercises, simplify, and write your answer in decimal form.

A helicopter flying at an altitude of 1000 feet drops a rescue package. The polynomial $−16 t^{2} + 1000$ gives the height of the package $t$ seconds a after it was dropped. Find the height when $t = 6$ seconds.

Answer

424 feet

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Problems and answers are adapted from Elementary Algebra 2e by Lynn Marecek and MaryAnne Anthony-Smith, OpenStax, and are used under the CC BY-NC-SA 4.0 licence. Access for free at https://openstax.org/books/elementary-algebra-2e/pages/1-introduction. Illustrations: Copyright Rice University, OpenStax, under CC BY-NC-SA 4.0 license. This adaptation is released under the same licence, is non-commercial, and is neither affiliated with nor endorsed by OpenStax or Rice University.