Chapter 1: PrerequisitesContent check — College Algebra, Chapter 1: Prerequisites

Write each of the following as a rational number.

ⓐ11

ⓑ3

ⓒ–4

Write each of the following rational numbers as either a terminating or repeating decimal.

ⓐ $\frac{68}{17}$

ⓑ $\frac{8}{13}$

ⓒ $-\frac{17}{20}$

Determine whether each of the following numbers is rational or irrational. If it is rational, determine whether it is a terminating or repeating decimal.

ⓐ $\frac{7}{77}$

ⓑ $\sqrt{81}$

ⓒ $4.27027002700027\dots$

ⓓ $\frac{91}{13}$

ⓔ $\sqrt{39}$

Classify each number as either positive or negative and as either rational or irrational. Does the number lie to the left or the right of 0 on the number line?

ⓐ $\sqrt{73}$

ⓑ $\mathrm{-11.411411411}\dots$

ⓒ $\frac{47}{19}$

ⓓ $-\frac{\sqrt{5}}{2}$

ⓔ $6.210735$

Classify each number as being a natural number (N), whole number (W), integer (I), rational number (Q), and/or irrational number (Q′).

ⓐ $-\frac{35}{7}$

ⓑ $0$

ⓒ $\sqrt{169}$

ⓓ $\sqrt{24}$

ⓔ $4.763763763\dots$

Use the order of operations to evaluate each of the following expressions.

ⓐ $\sqrt{5^2-4^2}+$ $7{\left(5-4\right)}^2$

ⓑ $1+$ $\frac{7\cdot 5-8\cdot 4}{9-6}$

ⓒ $|1.8-$ $4.3|+$ $0.4\sqrt{15+10}$

ⓓ $\frac{1}{2}[5\cdot 3^2-7^2]+$ $\frac{1}{3}\cdot 9^2$

ⓔ $[{\left(3-8\right)}^2-4]-$ $(3-8)$

Use the properties of real numbers to rewrite and simplify each expression. State which properties apply.

ⓐ $\left(-\frac{23}{5}\right)\cdot \left[11\cdot \left(-\frac{5}{23}\right)\right]$

ⓑ $5\cdot \left(6.2+0.4\right)$

ⓒ $18-\left(7\mathrm{-15}\right)$

ⓓ $\frac{17}{18}+$ $[\frac{4}{9}+(-\frac{17}{18}\left)\right]$

ⓔ $6\cdot \left(\mathrm{-3}\right)+6\cdot 3$

List the constants and variables for each algebraic expression.

ⓐ $2\pi r\left(r+h\right)$

ⓑ2(L + W)

ⓒ $4y^3+y$

Evaluate the expression $11-3y$ for each value for y.

ⓐ $y=2$

ⓑ $y=0$

ⓒ $y=\frac{2}{3}$

ⓓ $y=\mathrm{-5}$

Evaluate each expression for the given values.

ⓐ $\frac{y+3}{y-3}$ for $y=5$

ⓑ $7-2t$ for $t=\mathrm{-2}$

ⓒ $\frac{1}{3}\pi r^2$ for $r=11$

ⓓ ${\left(p^{2}q\right)}^3$ for $p=\mathrm{-2},q=3$

ⓔ $4(m-n)-$ $5(n-m)$ for $m=\frac{2}{3},n=\frac{1}{3}$

A photograph with length L and width W is placed in a mat of width 8 centimeters (cm). The area of the mat (in square centimeters, or cm²) is found to be $A=$ $(L+16)(W+16)-$ $L\cdot W.$ See Figure 4. Find the area of a mat for a photograph with length 32 cm and width 24 cm.

Simplify each algebraic expression.

ⓐ $\frac{2}{3}y-$ $2(\frac{4}{3}y+z)$

ⓑ $\frac{5}{t}-2-\frac{3}{t}+1$

ⓒ $4p(q-1)+$ $q(1-p)$

ⓓ $9r-$ $(s+2r)+$ $(6-s)$

For the following exercises, simplify the given expression.

$6÷2-\left(81÷3^2\right)$

For the following exercises, simplify the given expression.

$\mathrm{-2}\times {\left[16÷{\left(8-4\right)}^2\right]}^2$

For the following exercises, simplify the given expression.

$3\left(5-8\right)$

For the following exercises, simplify the given expression.

$12÷\left(36÷9\right)+6$

For the following exercises, simplify the given expression.

$3-12\times 2+19$

For the following exercises, simplify the given expression.

$5+\left(6+4\right)-11$

For the following exercises, simplify the given expression.

$14\times 3÷7-6$

For the following exercises, simplify the given expression.

$6+2\times 2-1$

For the following exercises, simplify the given expression.

$9+4\left(2^2\right)$

For the following exercises, simplify the given expression.

$25÷5^2-7$

For the following exercises, simplify the given expression.

$2\times 4-9\left(\mathrm{-1}\right)$

For the following exercises, simplify the given expression.

$12\left(3-1\right)÷6$

For the following exercises, evaluate the expression using the given value of the variable.

$4y+8–2y$ for $y=3$

For the following exercises, evaluate the expression using the given value of the variable.

$4z-$ $2z(1+4)–36$ for $z=5$

For the following exercises, evaluate the expression using the given value of the variable.

$-{\left(2x\right)}^2+1+3$ for $x=2$

For the following exercises, evaluate the expression using the given value of the variable.

$2\left(11c-4\right)–36$ for $c=0$

For the following exercises, evaluate the expression using the given value of the variable.

$\frac{1}{4}\left(8w-4^2\right)$ for $w=1$

For the following exercises, simplify the expression.

$2y-{\left(4\right)}^{2}y-11$

For the following exercises, simplify the expression.

$8b-4b\left(3\right)+1$

For the following exercises, simplify the expression.

$7z-3+z\times 6^2$

For the following exercises, simplify the expression.

$9\left(y+8\right)-27$

For the following exercises, simplify the expression.

$6+12b-3\times 6b$

For the following exercises, simplify the expression.

${\left(\frac{4}{9}\right)}^2\times 27x$

For the following exercises, simplify the expression.

$9x+$ $4x(2+3)-$ $4(2x+3x)$

For the following exercises, consider this scenario: Fred earns $40 at the community garden. He spends $10 on a streaming subscription, puts half of what is left in a savings account, and gets another $5 for walking his neighbor’s dog.

Write the expression that represents the number of dollars Fred keeps (and does not put in his savings account). Remember the order of operations.

For the following exercises, solve the given problem.

According to the U.S. Mint, the diameter of a quarter is 0.955 inches. The circumference of the quarter would be the diameter multiplied by $\pi .$ Is the circumference of a quarter a whole number, a rational number, or an irrational number?

For the following exercises, consider this scenario: There is a mound of $g$ pounds of gravel in a quarry. Throughout the day, 400 pounds of gravel is added to the mound. Two orders of 600 pounds are sold and the gravel is removed from the mound. At the end of the day, the mound has 1,200 pounds of gravel.

Write the equation that describes the situation.

For the following exercise, solve the given problem.

Ramon runs the marketing department at their company. Their department gets a budget every year, and every year, they must spend the entire budget without going over. If they spend less than the budget, then the department gets a smaller budget the following year. At the beginning of this year, Ramon got $2.5 million for the annual marketing budget. They must spend the budget such that $\mathrm{2,500,000}-$ $x=$ $0.$ What property of addition tells us what the value of x must be?

For the following exercises, use a graphing calculator to solve for x. Round the answers to the nearest hundredth.

${\left(0.25-0.75\right)}^{2}x-$ $7.2=$ $9.9$

For the following exercises, use a graphing calculator to solve for x. Round the answers to the nearest hundredth.

Determine whether the statement is true or false: The multiplicative inverse of a rational number is also rational.

For the following exercises, use a graphing calculator to solve for x. Round the answers to the nearest hundredth.

Determine whether the simplified expression is rational or irrational: $\sqrt{\mathrm{-18}-4\left(5\right)\left(\mathrm{-1}\right)}.$

For the following exercises, use a graphing calculator to solve for x. Round the answers to the nearest hundredth.

The division of two natural numbers will always result in what type of number?

Write each of the following products with a single base. Do not simplify further.

ⓐ $k^6\cdot k^9$

ⓑ ${\left(\frac{2}{y}\right)}^4\cdot \left(\frac{2}{y}\right)$

ⓒ $t^3\cdot t^6\cdot t^5$

Write each of the following products with a single base. Do not simplify further.

ⓐ $\frac{s^{75}}{s^{68}}$

ⓑ $\frac{{\left(\mathrm{-3}\right)}^6}{\mathrm{-3}}$

ⓒ $\frac{{\left(e{f}^2\right)}^5}{{\left(e{f}^2\right)}^3}$

Write each of the following products with a single base. Do not simplify further.

ⓐ ${\left({\left(3y\right)}^8\right)}^3$

ⓑ ${\left(t^5\right)}^7$

ⓒ ${\left({\left(-g\right)}^4\right)}^4$

Simplify each expression using the zero exponent rule of exponents.

ⓐ $\frac{t^7}{t^7}$

ⓑ $\frac{{\left(d{e}^2\right)}^{11}}{2{\left(d{e}^2\right)}^{11}}$

ⓒ $\frac{w^4\cdot w^2}{w^6}$

ⓓ $\frac{t^3\cdot t^4}{t^2\cdot t^5}$

Write each of the following quotients with a single base. Do not simplify further. Write answers with positive exponents.

ⓐ $\frac{{\left(\mathrm{-3}t\right)}^2}{{\left(\mathrm{-3}t\right)}^8}$

ⓑ $\frac{f^{47}}{f^{49}\cdot f}$

ⓒ $\frac{2k^4}{5k^7}$

Write each of the following products with a single base. Do not simplify further. Write answers with positive exponents.

ⓐ $t^{\mathrm{-11}}\cdot t^6$

ⓑ $\frac{{25}^{12}}{{25}^{13}}$

Simplify each of the following products as much as possible using the power of a product rule. Write answers with positive exponents.

ⓐ ${\left(g^2{h}^3\right)}^5$

ⓑ ${\left(5t\right)}^3$

ⓒ ${\left(\mathrm{-3}{y}^5\right)}^3$

ⓓ $\frac{1}{{\left(a^6{b}^7\right)}^3}$

ⓔ ${\left(r^3{s}^{\mathrm{-2}}\right)}^4$

Simplify each of the following quotients as much as possible using the power of a quotient rule. Write answers with positive exponents.

ⓐ ${\left(\frac{b^5}{c}\right)}^3$

ⓑ ${\left(\frac{5}{u^8}\right)}^4$

ⓒ ${\left(\frac{\mathrm{-1}}{w^3}\right)}^{35}$

ⓓ ${\left(p^{\mathrm{-4}}{q}^3\right)}^8$

ⓔ ${\left(c^{\mathrm{-5}}{d}^{\mathrm{-3}}\right)}^4$

Simplify each expression and write the answer with positive exponents only.

ⓐ ${\left(2u{v}^{-2}\right)}^{\mathrm{-3}}$

ⓑ $x^8\cdot x^{\mathrm{-12}}\cdot x$

ⓒ ${\left(\frac{e^2{f}^{-3}}{f^{\mathrm{-1}}}\right)}^2$

ⓓ $\left(9r^{\mathrm{-5}}{s}^3\right)\left(3r^6{s}^{\mathrm{-4}}\right)$

ⓔ ${\left(\frac{4}{9}t{w}^{\mathrm{-2}}\right)}^{\mathrm{-3}}{\left(\frac{4}{9}t{w}^{\mathrm{-2}}\right)}^3$

ⓕ $\frac{{\left(2h^{2}k\right)}^4}{{\left(7h^{\mathrm{-1}}{k}^2\right)}^2}$

Write each number in scientific notation.

ⓐU.S. national debt per taxpayer (April 2014): $152,000

ⓑWorld population (April 2014): 7,158,000,000

ⓒWorld gross national income (April 2014): $85,500,000,000,000

ⓓTime for light to travel 1 m: 0.00000000334 s

ⓔProbability of winning lottery (match 6 of 49 possible numbers): 0.0000000715

Convert each number in scientific notation to standard notation.

ⓐ $7.03\times {10}^5$

ⓑ $\mathrm{-8.16}\times {10}^{11}$

ⓒ $\mathrm{-3.9}\times {10}^{\mathrm{-13}}$

ⓓ $8\times {10}^{\mathrm{-6}}$

Perform the operations and write the answer in scientific notation.

ⓐ $\left(\mathrm{-7.5}\times {10}^8\right)\left(1.13\times {10}^{\mathrm{-2}}\right)$

ⓑ $\left(1.24\times {10}^{11}\right)÷\left(1.55\times {10}^{18}\right)$

ⓒ $\left(3.72\times {10}^9\right)\left(8\times {10}^3\right)$

ⓓ $\left(9.933\times {10}^{23}\right)÷\left(-2.31\times {10}^{17}\right)$

ⓔ $\left(\mathrm{-6.04}\times {10}^9\right)\left(7.3\times {10}^2\right)\left(\mathrm{-2.81}\times {10}^2\right)$

For the following exercises, simplify the given expression. Write answers with positive exponents.

$9^2$

For the following exercises, simplify the given expression. Write answers with positive exponents.

$3^2\times 3^3$

For the following exercises, simplify the given expression. Write answers with positive exponents.

${\left(2^2\right)}^{\mathrm{-2}}$

For the following exercises, simplify the given expression. Write answers with positive exponents.

${11}^3÷{11}^4$

For the following exercises, simplify the given expression. Write answers with positive exponents.

${\left(8^0\right)}^2$

For the following exercises, write each expression with a single base. Do not simplify further. Write answers with positive exponents.

$4^2\times 4^3÷4^{\mathrm{-4}}$

For the following exercises, write each expression with a single base. Do not simplify further. Write answers with positive exponents.

${\left({12}^3\times 12\right)}^{10}$

For the following exercises, write each expression with a single base. Do not simplify further. Write answers with positive exponents.

$7^{\mathrm{-6}}\times 7^{\mathrm{-3}}$

For the following exercises, express the decimal in scientific notation.

0.0000314

For the following exercises, convert each number in scientific notation to standard notation.

$1.6\times {10}^{10}$

For the following exercises, simplify the given expression. Write answers with positive exponents.

$\frac{a^3{a}^2}{a}$

For the following exercises, simplify the given expression. Write answers with positive exponents.

${\left(b^3{c}^4\right)}^2$

For the following exercises, simplify the given expression. Write answers with positive exponents.

$a{b}^2÷d^{\mathrm{-3}}$

For the following exercises, simplify the given expression. Write answers with positive exponents.

$\frac{m^4}{n^0}$

For the following exercises, simplify the given expression. Write answers with positive exponents.

$\frac{p^{\mathrm{-4}}{q}^2}{p^2{q}^{\mathrm{-3}}}$

For the following exercises, simplify the given expression. Write answers with positive exponents.

${\left(y^7\right)}^3÷x^{14}$

For the following exercises, simplify the given expression. Write answers with positive exponents.

$(25m)÷(5^{0}m)$

For the following exercises, simplify the given expression. Write answers with positive exponents.

$\frac{2^3}{{\left(3a\right)}^{\mathrm{-2}}}$

For the following exercises, simplify the given expression. Write answers with positive exponents.

${\left(b^{\mathrm{-3}}c\right)}^3$

For the following exercises, simplify the given expression. Write answers with positive exponents.

${\left(9z^3\right)}^{\mathrm{-2}}y$

For the following exercises, simplify the given expression. Write answers with positive exponents.

A dime is the thinnest coin in U.S. currency. A dime’s thickness measures $1.35\times {10}^{\mathrm{-3}}$ m. Rewrite the number in standard notation.

For the following exercises, simplify the given expression. Write answers with positive exponents.

A terabyte is made of approximately 1,099,500,000,000 bytes. Rewrite in scientific notation.

For the following exercises, simplify the given expression. Write answers with positive exponents.

One picometer is approximately $3.397\times {10}^{\mathrm{-11}}$ in. Rewrite this length using standard notation.

For the following exercises, use a graphing calculator to simplify. Round the answers to the nearest hundredth.

${\left(\frac{{12}^3{m}^{33}}{4^{\mathrm{-3}}}\right)}^2$

For the following exercises, simplify the given expression. Write answers with positive exponents.

${\left(\frac{3^2}{a^3}\right)}^{\mathrm{-2}}{\left(\frac{a^4}{2^2}\right)}^2$

For the following exercises, simplify the given expression. Write answers with positive exponents.

$\frac{m^2{n}^3}{a^2{c}^{\mathrm{-3}}}\cdot \frac{a^{\mathrm{-7}}{n}^{\mathrm{-2}}}{m^2{c}^4}$

For the following exercises, simplify the given expression. Write answers with positive exponents.

${\left(\frac{{\left(a{b}^{2}c\right)}^{\mathrm{-3}}}{b^{\mathrm{-3}}}\right)}^2$

For the following exercises, simplify the given expression. Write answers with positive exponents.

Planck’s constant is an important unit of measure in quantum physics. It describes the relationship between energy and frequency. The constant is written as $6.62606957\times {10}^{\mathrm{-34}}.$ Write Planck’s constant in standard notation.

Evaluate each expression.

ⓐ $\sqrt{225}$

ⓑ $\sqrt{\sqrt{81}}$

ⓒ $\sqrt{25-9}$

ⓓ $\sqrt{36}+\sqrt{121}$

Simplify.

ⓐ $\sqrt[3]{\mathrm{-216}}$

ⓑ $\frac{3\sqrt[4]{80}}{\sqrt[4]{5}}$

ⓒ $6\sqrt[3]{9,000}+$ $7\sqrt[3]{576}$

For the following exercises, simplify each expression.

$\sqrt{256}$

For the following exercises, simplify each expression.

$\sqrt{4\left(9+16\right)}$

For the following exercises, simplify each expression.

$\sqrt{196}$

For the following exercises, simplify each expression.

$\sqrt{98}$

For the following exercises, simplify each expression.

$\sqrt{\frac{81}{5}}$

For the following exercises, simplify each expression.

$\sqrt{169}+\sqrt{144}$

For the following exercises, simplify each expression.

$\frac{18}{\sqrt{162}}$

For the following exercises, simplify each expression.

$14\sqrt{6}-6\sqrt{24}$

For the following exercises, simplify each expression.

$\sqrt{150}$

For the following exercises, simplify each expression.

$\left(\sqrt{42}\right)\left(\sqrt{30}\right)$

For the following exercises, simplify each expression.

$\sqrt{\frac{4}{225}}$

For the following exercises, simplify each expression.

$\sqrt{\frac{360}{361}}$

For the following exercises, simplify each expression.

$\frac{8}{1-\sqrt{17}}$

For the following exercises, simplify each expression.

$\sqrt[3]{128}+3\sqrt[3]{2}$

For the following exercises, simplify each expression.

$\frac{15\sqrt[4]{125}}{\sqrt[4]{5}}$

For the following exercises, simplify each expression.

$\sqrt{400x^4}$

For the following exercises, simplify each expression.

$\sqrt{49p}$

For the following exercises, simplify each expression.

$m^{\frac{5}{2}}\sqrt{289}$

For the following exercises, simplify each expression.

$3\sqrt{a{b}^2}-b\sqrt{a}$

For the following exercises, simplify each expression.

$\sqrt{\frac{225x^3}{49x}}$

For the following exercises, simplify each expression.

$\sqrt{50y^8}$

For the following exercises, simplify each expression.

$\sqrt{\frac{32}{14d}}$

For the following exercises, simplify each expression.

$\frac{\sqrt{8}}{1-\sqrt{3x}}$

For the following exercises, simplify each expression.

$w^{\frac{3}{2}}\sqrt{32}-w^{\frac{3}{2}}\sqrt{50}$

For the following exercises, simplify each expression.

$\frac{\sqrt{12x}}{2+2\sqrt{3}}$

For the following exercises, simplify each expression.

$\sqrt{125n^{10}}$

For the following exercises, simplify each expression.

$\sqrt{\frac{81m}{361m^2}}$

For the following exercises, simplify each expression.

$\sqrt{\frac{144}{324d^2}}$

For the following exercises, simplify each expression.

$\sqrt[4]{\frac{162x^6}{16x^4}}$

For the following exercises, simplify each expression.

$\sqrt[3]{128z^3}-$ $\sqrt[3]{\mathrm{-16}{z}^3}$

For the following exercises, simplify each expression.

A guy wire for a suspension bridge runs from the ground diagonally to the top of the closest pylon to make a triangle. We can use the Pythagorean Theorem to find the length of guy wire needed. The square of the distance between the wire on the ground and the pylon on the ground is 90,000 feet. The square of the height of the pylon is 160,000 feet. So the length of the guy wire can be found by evaluating $\sqrt{\mathrm{90,000}+\mathrm{160,000}}.$ What is the length of the guy wire?

For the following exercises, simplify each expression.

$\frac{\sqrt{8}-\sqrt{16}}{4-\sqrt{2}}-2^{\frac{1}{2}}$

For the following exercises, simplify each expression.

$\frac{\sqrt{m{n}^3}}{a^2\sqrt{c^{\mathrm{-3}}}}\cdot \frac{a^{\mathrm{-7}}{n}^{\mathrm{-2}}}{\sqrt{m^2{c}^4}}$

For the following exercises, simplify each expression.

$\frac{x\sqrt{64y}+4\sqrt{y}}{\sqrt{128y}}$

For the following exercises, simplify each expression.

$\sqrt{\frac{\sqrt[3]{64}+\sqrt[4]{256}}{\sqrt{64}+\sqrt{256}}}$

Find the sum.

$(2x^3+5x^2-x+1)+$ $(2x^2-3x-4)$

Find the difference.

$(\mathrm{-7}{x}^3-7x^2+6x-2)-$ $(4x^3-6x^2-x+7)$

Find the product.

$(3x+2)\left(x^3-4x^2+7\right)$

Use FOIL to find the product.

$(x+7)(3x-5)$

For the following exercises, identify the degree of the polynomial.

$7x-2x^2+13$

For the following exercises, identify the degree of the polynomial.

$\mathrm{-625}{a}^8+16b^4$

For the following exercises, identify the degree of the polynomial.

$x^2+4x+4$

For the following exercises, find the sum or difference.

$(12x^2+3x)-$ $\left(8x^2\mathrm{-19}\right)$

For the following exercises, find the sum or difference.

$(6w^2+24w+24)-$ $(3w-26w+3)$

For the following exercises, find the sum or difference.

$(11b^4-6b^3+18b^2-4b+8)-$ $(3b^3+6b^2+3b)$

For the following exercises, find the product.

$(4x+2)(6x-4)$

For the following exercises, find the product.

$\left(6b^2-6\right)\left(4b^2-4\right)$

For the following exercises, find the product.

$(9v-11)(11v-9)$

For the following exercises, find the product.

$(8n-4)\left(n^2+9\right)$

For the following exercises, expand the binomial.

${(3y-7)}^2$

For the following exercises, expand the binomial.

${\left(4p+9\right)}^2$