Chapter 3: Real Number Systems and Number TheoryContent check — Contemporary Math, Chapter 3: Real Number Systems and Number Theory

You can test if the number is divisible by 9. Add the digits together. If that sum is divisible by 9, then so is the original number.

5 + 4 = 9, so 54 is divisible by 9.

You only need to check numbers up to the square root of 1,429, which is approximately 37.80.

Step 1: Use the known rules of divisibility.

2: The last digit is even.

The last digit is not even.

3: Add the digits of the number together. If that sum is divisible by 3, then so is the original number.

1 + 4 + 2 + 9 = 16

16 is not divisible by 3.

5: If the last digit is 5 or 0, then the original number is divisible by 5.

The last digit is not 5 or 0, so the number is not divisible by 5.

Step 2:

Use a calculator to test the primes up to 37.

7, 11, 13, 17, 19, 23, 29, 31, 37

For instance, 1,429 ÷ 7 ≈ 178.4285714.

None of these primes result in an integer quotient. They all have a decimal part.

The conclusion is that 1,429 is a prime number.

You only need to check numbers up to the square root of 859, which is approximately 29.308.

Step 1: Use the known rules of divisibility.

2: The last digit is even.

The last digit is not even.

3: Add the digits of the number together. If that sum is divisible by 3, then so is the original number.

8 + 5 = 13

13 is not divisible by 3.

5: If the last digit is 5 or 0, then the original number is divisible by 5.

The last digit is not 5 or 0, so the number is not divisible by 5.

Step 2:

Use a calculator to test the primes up to 29.

7, 11, 13, 17, 19, 23, 29

For instance, 859 ÷ 7 ≈ 122.7142857.

None of these primes result in an integer quotient. They all have a decimal part.

The conclusion is that 859 is a prime number.

You only need to check numbers up to the square root of 5,067,322, which is approximately 2251.071301.

Step 1: Use the known rules of divisibility.

2: The last digit is even.

The last digit is even.

5,067,322 = 2 × 2,533,661

The conclusion is that 5,067,322 is a composite number.

You only need to check numbers up to the square root of 1,477, which is approximately 38.4317577.

Step 1: Use the known rules of divisibility.

2: The last digit is even.

The last digit is not even.

3: Add the digits of the number together. If that sum is divisible by 3, then so is the original number.

1 + 4 + 7 + 7 = 19

19 is not divisible by 3.

5: If the last digit is 5 or 0, then the original number is divisible by 5.

The last digit is not 5 or 0, so the number is not divisible by 5.

Step 2:

Use a calculator to test the primes up to 38.

7, 11, 13, 17, 19, 23, 29, 31, 37

For instance, 1,477 ÷ 7 = 211.

7 is an integer factor, so the conclusion is that 1,477 is a composite number.

Use the divisibility rules for the primes, starting from the smallest: 2, 3, 5, 7, 11, 13, 17, 19.

Because 90 is even, you know it is divisible by 2.

$90=2\times 45$

Because 45 ends in 5, you know it is divisible by 5.

$90=2\times 5\times 9$

$90=2\times 5\times 3\times 3$

$90=2\times 3^2\times 5$

You know 85 has a factor of 5 because it ends in 5.

$85=5\times 17$

Because 17 is a prime number, you are done.

Because 280 ends in 0, you know it has a factor of 10, which is 2 times 5. Factor out 2 and 5.

$280=2\times 5\times 28$

28 is divisible by 2, so factor out 2 again.

$280=2\times 2\times 5\times 14$

14 is even, so factor out 2 again.

$280=2\times 2\times 2\times 5\times 7$

Or you can write the factorization: $280=2^3\times 5\times 7$.

Because 180 ends in 0, you know it has a factor of 10, which is 2 times 5. Factor out 2 and 5.

$180=2\times 5\times 18$

18 is divisible by 2, so factor out 2 again.

$180=2\times 2\times 5\times 9$

Factor 9.

$180=2\times 2\times 5\times 3\times 3$

180 has three prime factors (2, 3, and 5).

You need the GCD of the width and length: 400 cm by 540 cm

Greatest common divisor = $2\times 2\times 5=20\mathrm{c}\mathrm{m}$

The largest square bricks that can be used are 20 cm by 20 cm.

Find the GCD of 21, 35, and 28.

Greatest common divisor = 7

The largest team size is 7 students.

Multiples of 12: 12, 24, 36, 48, 60

Multiples of 15: 15, 30, 45, 60

The first number common to both lists is 60.

The LCM is 60.

Find the prime factorization of each number.

$20=2^2\times 5$

$28=2^2\times 7$

Make a table of each prime and the largest exponent for each prime.

The LCM is the product of each prime raised to the powers identified in the table.

The LCM is $2^2\times 5\times 7=140$.

Find the prime factorization of each number.

$150=2\times 3\times 5^2$

$240=2^4\times 3\times 5$

$462=2\times 3\times 7\times 11$

Make a table of each prime and the largest exponent for each prime.

The LCM is the product of each prime raised to the powers identified in the table.

The LCM is $2^4\times 3\times 5^2\times 7\times 11=92,400$.

Using lists:

List multiples of each number until you find a common number.

18: 18, 36, 54, 72, 90, 108, 126, 144, 162, 180, 198, 216, 234, 252, 270, 288, 306, 324, 342, 360

24: 24, 48, 72, 120, 144, 168, 192, 216, 240, 264, 288, 312, 336, 360

40: 40, 80, 120, 160, 200, 240, 280, 320, 360

The first common number is 360. The LCM is 360.

Using prime factorization:

Find the prime factorization of each number.

$18=2\times 3^2$

$24=2^3\times 3$

$40=2^3\times 5$

Make a table of each prime and the largest exponent for each prime.

The LCM is the product of each prime raised to the powers identified in the table.

The LCM is $2^3\times 3^2\times 5=360$.

Find the LCM of 255 and 4,330.

Find the prime factorization of each number.

$255=3\times 5\times 17$

$4,330=2\times 5\times 433$

Make a table of each prime and the largest exponent for each prime.

The LCM is the product of each prime raised to the powers identified in the table.

The LCM i $2\times 3\times 5\times 17\times 433=220,830$.

The sun, Venus, and Jupiter will align in 220,830 days.

Find the LCM of 130 and 900.

Find the prime factorization of each number.

$255=2\times 5\times 13$

$900=2^2\times 3^2\times 5^2$

Make a table of each prime and the largest exponent for each prime.

The LCM is the product of each prime raised to the powers identified in the table.

The LCM is $2^2\times 3^2\times 5^2\times 13=11,700$.

The prize winner is the one who submits the 11,700th submission.

31 and 701 are prime.

The rest are composite: 56, 213, 48.

$56=2\times 2\times 2\times 7$

$213=3\times 71$

$48=2\times 2\times 2\times 2\times 3$

4,570 ends in 0, so it has a factor of 10. Divide out 2 and 5.

$4,570=2\times 5\times 457$

The square root of 457 is approximately 21.37.

457 is not divisible by 2, 3, 5, 7, 11, 13, 17, or 19, the primes less than 21.

Thus, you are done and $4,570=2\times 5\times 457$.

The greatest common divisor is the product of the common factors. There is only one common factor: 2.

Find the LCM of 45 and 70.

Find the prime factorization of each number.

$45=3^2\times 5$

$70=2\times 5\times 7$

Make a table of each prime and the largest exponent for each prime.

The LCM is the product of each prime raised to the powers identified in the table.

The LCM is $2\times 3^2\times 5\times 7=630$.

Find the GCD of 30, 20, and 70.

Greatest common divisor = $2\times 5=10$

The maximum number of bags is 10.

Consider a number line with one tick mark for every integer. To graph –38, you move 38 tick marks to the left of zero. To graph 27, you move 27 tick marks to the right of zero.

–38 is farther left than 27, so –38 is less than 27.

$–38<27$

$27>–38$

Consider a number line with one tick mark for every integer. To graph –213, you move 213 tick marks to the left of zero. To graph –63, you move 63 tick marks to the left of zero.

–213 is farther left than –63, so –213 is less than –63.

$–213<–63$

$–63>–213$

Because the value inside the absolute value is positive, the absolute value is just the number itself.

$|38|=38$

Because the value inside the absolute value bars is negative, the absolute value removes the negative sign.

|–81| = 81

Use your calculator to find that 38 – 100 is –62.

Because 100 is larger than 38, you expect the difference to be negative.

Use your calculator to find that 19 + (–36) is –17.

Because you are adding numbers with opposite signs, the sign of the answer matches the sign of the integer with the larger absolute value. Because |–36| > |19|, the sign of the answer matches the sign of –36.

Chanel owes Chrisian money, so represent $180 as a positive number. Christian owes Jeff money, so subtract 91 from 180.

$180-91=89$

Christian’s net worth is $89.

$81\times 26\mathrm{i}\mathrm{s}2,106$.

Because 81 and 26 are both positive, the product is positive.

$\left(-18\right)\times 13\mathrm{i}\mathrm{s}-234$.

Because –18 and 13 have opposite signs, the product is negative.

$\left(-116\right)÷4\mathrm{i}\mathrm{s}-29$.

Because –116 and 4 have opposite signs, the quotient is negative.

$\left(-77\right)÷\left(-11\right)\mathrm{i}\mathrm{s}7$.

Because –77 and –11 have the same sign, the quotient is positive.

Add the seven account balances, then divide by 7.

Sum: $1250+673+\left(-1500\right)+1000+785+785+710$

Divide the sum by 7:

$3,703÷7=529$

The average balance is $529.

–4 is an integer because it is the negative of a counting number.

15.2 is not an integer as it has a decimal part. It is between two integers, 15 and 16.

The square root of 2 is not an integer. It is approximately 1.414, which is between two integers, 1 and 2.

The fraction 3 divided by 20 is not an integer. It is equal to 0.15, which is between two integers, 1 and 2.

430 is an integer. It is a counting number.

Use a number line marked with tick marks to represent every integer.

4: Place a solid dot four tick marks to the right of zero.

–2: Place a solid dot two tick marks to the left of zero.

7: Place a solid dot seven tick marks to the right of zero.

Consider a number line marked with tick marks to represent every integer. The farther a number is to the left, the less its value.

The negative numbers are on the left of zero.

The number farthest to the left is –13, then –7, then –2.

The positive numbers are to the right of zero.

Of the positive numbers, 4 is left of 10.

In increasing order: –13, –7, –2, 4, 10

Because the value inside the absolute value bars is negative, the absolute value removes the negative sign.

$\left|-7\right|=7$

$\left(-3\right)\times \left(-12\right)=36$

Because –3 and –12 have the same sign, the product is positive.

Section 3.2 Exercises

$43+18\times 15$

Use PEMDAS. The highest order operation in this expression is multiplication.

${60}^2/50$

Use PEMDAS.

$3\times 6^3-18$

Use PEMDAS.

$13+3^4+8/6+5^2-3\times 4$

Use PEMDAS.

$12/\left(-4\right)+8\times 9/12\times 2^3-24\times 25/10$

Use PEMDAS.

$3\times 4^3\times 7+24/6\times 7^2-9/3\times 8$

Use PEMDAS.

$8-\left(25-2^2\right)/7$

Use PEMDAS.

Evaluate the operations inside the parentheses first.

$(8-6{)}^2\times 100-({(48/6-3)}^2-4\times 7)$

Use PEMDAS.

Evaluate the operations inside the parentheses first.

Parentheses have the highest precedence.

Order of Operations (PEMDAS)

Parentheses

Exponents

Multiplication/Division

Addition/Subtraction

For same order operations, work left to right.

Exponents are performed before addition.

Order of Operations (PEMDAS)

Parentheses

Exponents

Multiplication/Division

Addition/Subtraction

For same order operations, work left to right.

$2\times 3^2-5\times 8$

Use PEMDAS.

$(4-3{)}^2+27\times 8^2÷6^2$

Use PEMDAS.

94 is not a perfect square.

The prime factorization of 94 is $2\times 47$. None of the factors is a pair.

Not rational

$\sqrt{13}\approx \mathrm{3.60555....}$. There is no repeating pattern, so this is not a rational number.

Two fractions, $\frac{a}{b}$ and $\frac{c}{d}$ are equivalent if $a\times d=b\times c$.

$a\times d=8\times 26=208$

$b\times c=14\times 12=168$

Because the products are not the same, the fractions are not equivalent.

Enter the fractions in Desmos (desmos.com) using “/” for the fraction bar.

What you enter: $124/297$

You see a fraction in the workspace and a decimal in the answer area.

Arrow out of the fraction and press $+.$

Enter the second fraction: $3/125$

The answer area shows $\mathrm{0.4415084175.}$

Click on the “convert to fraction button.” It looks like $\frac{◻}{◻}$.

You will see the final answer, $\frac{16,391}{37,125}$.

$\frac{38}{73}+\frac{7}{73}$

When two fractions have the same denominator, add the numerators.

$\frac{38+7}{73}$

$\frac{45}{73}$

There are no common factors to divide out.

$\frac{21}{40}-\frac{8}{40}$

When two fractions have the same denominator, subtract the numerators.

$\frac{21-8}{40}$

$\frac{13}{40}$

There are no common factors to divide out.

$\frac{4}{9}+\frac{7}{12}$

Find the LCM of 9 and 12. (You can use Desmos to do this.)

$\mathrm{L}\mathrm{C}\mathrm{M}=2\times 2\times 3\times 3=36$

Rewrite the fractions with 36 as the denominator.

$\frac{4\times 4}{9\times 4}+\frac{7\times 3}{12\times 3}$

$\frac{16}{36}+\frac{21}{36}$

Now, you can add the fractions by adding the numerators.

$\frac{37}{36}$

There are no common factors to divide out.

Alternative answer: $1\frac{1}{36}$

$\frac{10}{99}-\frac{17}{300}$

Find the LCM of 99 and 300. (You can use Desmos to do this.)

$\mathrm{L}\mathrm{C}\mathrm{M}=2\times 2\times 3\times 3\times 5\times 5\times 11=9,900$

Rewrite the fractions with 9,900 as the denominator.

$\frac{10\times 100}{99\times 100}-\frac{17\times 33}{300\times 33}$

$\frac{1,000}{9,900}-\frac{561}{9,900}$

Now, you can subtract the fractions by subtracting the numerators.

$\frac{1,000-561}{9,900}$

$\frac{439}{9,900}$

There are no common factors to divide out.

$95÷26$ is 3 with a remainder. What is that remainder?

$26\times 3=78$

$95-78=17$

The remainder is 17.

So, write $\frac{95}{26}$ as $3\frac{17}{26}$.

$9\frac{5}{14}$

The third decimal digit is 8. The next digit is 2, so round down.

5.108

The second decimal digit is 2. The next digit is 9, so round up. Change the 2 to 3.

18.63

Use a calculator to divide 48 by 30 to get 1.6.

Or because you can simplify to have 10 in the denominator, you can also find the decimal this way.

$\frac{48}{30}=\frac{3\times 16}{3\times 10}$

$\frac{16}{10}=1.6$

There are five decimal places, so raise 10 to the fifth power.

${10}^5=100,000$

$\frac{17.03347\times 100,000}{1\times 100,000}=\frac{1,703,347}{100,000}$

$\frac{45}{88}\times \frac{28}{75}$

Multiply the numerators and place that in the numerator.

Multiply the denominators and place that in the denominator.

$\frac{45\times 28}{88\times 75}$

$\frac{1,260}{6,600}$

Simplify. The GCF of 1,260 and 6,600 is 60.

$\frac{60\times 21}{60\times 110}$

$\frac{21}{110}$

$45.63÷17.13$ is approximately 2.663747 . . . .

The third decimal place is 3. The next digit is 7, so round up. Change the 3 to 4.

2.664

$\frac{46}{175}÷\frac{69}{285}$

Multiply the first fraction by the reciprocal of the second fraction.

$\frac{46}{175}\times \frac{285}{69}=\frac{13,110}{12,075}$

Simplify.

$\frac{38\times 345}{35\times 345}$

$\frac{38}{35}$

$\frac{3}{40}÷\frac{42}{55}$

Multiply the first fraction by the reciprocal of the second fraction.

$\frac{3}{40}\times \frac{55}{42}=\frac{165}{1,680}$

Simplify.

$\frac{11\times 15}{112\times 15}$

$\frac{11}{112}$

${\left(\frac{3}{16}+\frac{7}{16}\right)}^2+\frac{1}{5}÷\frac{3}{10}$

Use PEMDAS.

$\left(\frac{3}{5}+2\right)\times {\left(\frac{4}{5}-\frac{1}{2}\right)}^2÷\frac{11}{15}$

Use PEMDAS.

Optional decimal form: $0.31\overline{90}$

The density property process:

Step 1: Add the two rational numbers.

$\frac{27}{13}+\frac{21}{10}=\frac{27\times 10}{13\times 10}+\frac{21\times 13}{10\times 13}$

$\frac{270}{130}+\frac{273}{130}$

$\frac{543}{130}$

Step 2: Divide that result by 2. Dividing by 2 is the same as multiplying by one-half.

$\frac{543}{130}\times \frac{1}{2}=\frac{543}{260}$

There are other numbers, but this is the one found using this process.

Math: half the $\mathrm{t}\mathrm{i}\mathrm{m}\mathrm{e}=\frac{1}{2}\times 10=5\mathrm{h}\mathrm{o}\mathrm{u}\mathrm{r}\mathrm{s}$ on math

History: a quarter of the $\mathrm{t}\mathrm{i}\mathrm{m}\mathrm{e}=\frac{1}{4}\times 10=2.5\mathrm{h}\mathrm{o}\mathrm{u}\mathrm{r}\mathrm{s}$ on history

Writing: an eighth of the $\mathrm{t}\mathrm{i}\mathrm{m}\mathrm{e}=\frac{1}{8}\times 10=1.25\mathrm{h}\mathrm{o}\mathrm{u}\mathrm{r}\mathrm{s}$ on writing

Physics: an eighth of the $\mathrm{t}\mathrm{i}\mathrm{m}\mathrm{e}=\frac{1}{8}\times 10=1.25\mathrm{h}\mathrm{o}\mathrm{u}\mathrm{r}\mathrm{s}$ on physics

30% of 2,400 calories

$\frac{30}{100}\times 2,400=720$

Callum should consume 720 calories of protein.

A percent is a specific rational number and is literally per 100. $n$ percent, denoted $n\mathrm{\%}$, is the fraction $\frac{n}{100}.$

$4\mathrm{\%}=\frac{4}{100}$

Alternate correct answer: $\frac{1}{25}$

A percent is a specific rational number and is literally per 100. $n$ percent, denoted $n\mathrm{\%}$, is the fraction $\frac{n}{100}.$

$50\mathrm{\%}=\frac{50}{100}$

Alternate correct answer: $\frac{1}{2}$

A percent is a specific rational number and is literally per 100. $n$ percent, denoted $n\mathrm{\%}$, is the fraction $\frac{n}{100}.$

$14\mathrm{\%}=\frac{14}{100}$

In decimal form: 0.14

A percent is a specific rational number and is literally per 100. $n$ percent, denoted $n\mathrm{\%}$, is the fraction $\frac{n}{100}.$

$7\mathrm{\%}=\frac{7}{100}$

In decimal form: 0.07

25% of 1,200

53% of 1,588

25% of the total is 30.

45% of the total is 360.

$n\mathrm{\%}$ of $x$ items is $\frac{n}{100}\times x$.

$n$ percent, denoted $n\mathrm{\%}$, is the fraction $\frac{n}{100}.$

70 is what percentage of 1,000?

Rewrite “is” as equals and “percentage of” as $\frac{n}{100}\times 1,000.$

$70=\frac{n}{100}\times 1,000$

$7=n$

The answer is 7%.

$n$ percent, denoted $n\mathrm{\%}$, is the fraction $\frac{n}{100}.$

425 is what percentage of 500?

Rewrite “is” as equals and “percentage of” as $\frac{n}{100}\times 500.$

$425=\frac{n}{100}\times 500$

$85=n$

The answer is 85%.

What is 20% of 2,200?

$n$ percent, denoted $n\mathrm{\%}$, is the fraction $\frac{n}{100}.$

Rily should eat 440 calories of protein.

$n$ percent, denoted $n\mathrm{\%}$, is the fraction $\frac{n}{100}.$

54 is what percentage of 450?

Rewrite “is” as equals and “percentage of” as $\frac{n}{100}\times 500.$

$54=\frac{n}{100}\times 450$

$12=n$

The answer is 12% of registered voters voted in the primaries.

40% of the total is $30.

$n\mathrm{\%}$ of $x$ items is $\frac{n}{100}\times x$.

The original price was $75.

$-41$ is a rational number as it is an integer. You can write an integer as a fraction over 1.

$\frac{4}{3}$ is a rational number as it is a ratio of two integers.

2.75 is a rational number as it equals a ratio of two integers. For instance, you can see it is equal to 275 divided by 100.

$0.2\overline{13}$ is a rational number because repeating decimals are rational numbers.

The square root of 13 is not a rational number because it approximates to a decimal that does not repeat.

$\frac{3}{8}+\frac{5}{12}$

What is the LCM of 8 and 12?

$\mathrm{L}\mathrm{C}\mathrm{M}=2\times 2\times 2\times 3=24$

The LCM of 8 and 12 is 24.

Rewrite both fractions with 24 as a denominator.

$\frac{3\times 3}{8\times 3}+\frac{5\times 2}{12\times 2}$

$\frac{9}{24}+\frac{10}{24}$

Add the numerators now that the denominators are the same.

$\frac{19}{24}$

Step 1: Because 0.34 has two decimal places, write 10 to the second power: 100.

Step 2: Remove the decimal point from the original number: 34.

Step 3: Write a fraction with the result from Step 2 in the numerator and the result from Step 1 in the denominator.

$\frac{34}{100}$

Alternate answer in lowest terms: $\frac{17}{50}$

$47÷12$ is 3 with a remainder. What is that remainder?

$12\times 3=36$

$47-36=11$

The remainder is 11.

So, $\frac{47}{36}$ as a mixed number is $3\frac{11}{12}$.

Multiply the numerators and denominators, dividing out common factors.

$\frac{2\times 21}{9\times 22}$

$\frac{2\times 7\times 3}{3\times 3\times 2\times 11}$

$\frac{7}{33}$

$\frac{2}{5}÷\frac{3}{10}+\frac{1}{6}$

Use PEMDAS. Do the division first by multiplying by the reciprocal.

$\frac{2}{5}\times \frac{10}{3}+\frac{1}{6}$

Multiply the numerators and denominators, dividing out common factors.

$\frac{2}{5}\times \frac{2\times 5}{3}+\frac{1}{6}$

$\frac{4}{3}+\frac{1}{6}$

Rewrite the first fraction to have 6 in the denominator.

$\frac{4\times 2}{3\times 2}+\frac{1}{6}$

$\frac{8}{6}+\frac{1}{6}$

Add the numerators.

$\frac{9}{6}$

Divide out common factors.

$\frac{3\times 3}{2\times 3}$

$\frac{3}{2}$

You can use the density property process:

Step 1: Add the two rational numbers.

$\frac{7}{8}+\frac{20}{21}=\frac{7\times 21}{8\times 21}+\frac{20\times 8}{21\times 8}$

$\frac{147}{168}+\frac{160}{168}$

$\frac{307}{168}$

Step 2: Divide that result by 2. Dividing by 2 is the same as multiplying by one-half.

$\frac{307}{168}\times \frac{1}{2}=\frac{307}{336}$

There are other numbers, but this is the one found using this process.

$3\frac{2}{7}$

Find one-eighth of $882.

$\frac{1}{8}\times 882=110.25$

Divide 882 by 8.

Lina will save $110.25.

What is 38% of 600?

$n$ percent, denoted $n\mathrm{\%}$, is the fraction $\frac{n}{100}.$

What is 10% of 70?

$n$ percent, denoted $n\mathrm{\%}$, is the fraction $\frac{n}{100}.$

The company will hire 7 new employees.

This is a perfect square.

$\frac{12}{221}=\frac{3\times 4}{13\times 17}$

This is not a perfect square as you do not have pairs of prime factors.

$\sqrt{225}=15$

This is not irrational because you can write 15 as the ratio of two integers.

$\sqrt{550}=\sqrt{25}\sqrt{22}=5\sqrt{22}$

The rational part is 5.

The irrational part is $\sqrt{22}$.

$\sqrt{773}$

773 is a prime number, so this cannot be further simplified.

The rational part is 1. (You can always have a factor of 1.)

The irrational part is $\sqrt{15}$.

$\sqrt{1,815}=\sqrt{121}\sqrt{15}=11\sqrt{15}$

The rational part is 11.

The irrational part is $\sqrt{15}$.

$41\sqrt{15}-23\sqrt{15}$

Because they have the same irrational part, you can subtract without using a calculator.

The formula uses the distributive property: $\left(a\times x\right)\pm \left(b\times x\right)=\left(a\pm b\right)\times x$.

$\left(41-23\right)\times \sqrt{15}$ Subtract.

$18\sqrt{15}$

$4.1\pi +3.2\pi$

Because they have the same irrational part, you can add without using a calculator.

The formula uses the distributive property: $\left(a\times x\right)\pm \left(b\times x\right)=\left(a\pm b\right)\times x$.