Chapter 3

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.

$90=2\times <!--\; \times \; -->45$

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

$90=2\times <!--\; \times \; -->5\times <!--\; \times \; -->9$

$90=2\times <!--\; \times \; -->5\times <!--\; \times \; -->3\times <!--\; \times \; -->3$

$90=2\times <!--\; \times \; -->3^2\times <!--\; \times \; -->5$

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

$85=5\times <!--\; \times \; -->17$

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.

$280=2\times <!--\; \times \; -->5\times <!--\; \times \; -->28$

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

$280=2\times <!--\; \times \; -->2\times <!--\; \times \; -->5\times <!--\; \times \; -->14$

14 is even, so factor out 2 again.

$280=2\times <!--\; \times \; -->2\times <!--\; \times \; -->2\times <!--\; \times \; -->5\times <!--\; \times \; -->7$

Or you can write the factorization: $280=2^3\times <!--\; \times \; -->5\times <!--\; \times \; -->7$.

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

$180=2\times <!--\; \times \; -->5\times <!--\; \times \; -->18$

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

$180=2\times <!--\; \times \; -->2\times <!--\; \times \; -->5\times <!--\; \times \; -->9$

Factor 9.

$180=2\times <!--\; \times \; -->2\times <!--\; \times \; -->5\times <!--\; \times \; -->3\times <!--\; \times \; -->3$

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 = $2\times <!--\; \times \; -->2\times <!--\; \times \; -->5=20\mathrm{c}\mathrm{m}$

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.

$20=2^2\times <!--\; \times \; -->5$

$28=2^2\times <!--\; \times \; -->7$

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 $2^2\times <!--\; \times \; -->5\times <!--\; \times \; -->7=140$.

Find the prime factorization of each number.

$150=2\times <!--\; \times \; -->3\times <!--\; \times \; -->5^2$

$240=2^4\times <!--\; \times \; -->3\times <!--\; \times \; -->5$

$462=2\times <!--\; \times \; -->3\times <!--\; \times \; -->7\times <!--\; \times \; -->11$

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 $2^4\times <!--\; \times \; -->3\times <!--\; \times \; -->5^2\times <!--\; \times \; -->7\times <!--\; \times \; -->11=92,400$.

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.

$18=2\times <!--\; \times \; -->3^2$

$24=2^3\times <!--\; \times \; -->3$

$40=2^3\times <!--\; \times \; -->5$

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 $2^3\times <!--\; \times \; -->3^2\times <!--\; \times \; -->5=360$.

Find the LCM of 255 and 4,330.

Find the prime factorization of each number.

$255=3\times <!--\; \times \; -->5\times <!--\; \times \; -->17$

$4,330=2\times <!--\; \times \; -->5\times <!--\; \times \; -->433$

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 $2\times <!--\; \times \; -->3\times <!--\; \times \; -->5\times <!--\; \times \; -->17\times <!--\; \times \; -->433=220,830$.

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.

$255=2\times <!--\; \times \; -->5\times <!--\; \times \; -->13$

$900=2^2\times <!--\; \times \; -->3^2\times <!--\; \times \; -->5^2$

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 $2^2\times <!--\; \times \; -->3^2\times <!--\; \times \; -->5^2\times <!--\; \times \; -->13=11,700$.

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

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.

$27>–38$

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.

$–63>–213$

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.

$81\times <!--\; \times \; -->26\mathrm{i}\mathrm{s}2,106$.

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

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

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.

$43+18\times <!--\; \times \; -->15$

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

${60}^2/50$

Use PEMDAS.

$3\times <!--\; \times \; -->6^3-<!--\; -\; -->18$

Use PEMDAS.

$13+3^4+8/6+5^2-<!--\; -\; -->3\times <!--\; \times \; -->4$

Use PEMDAS.

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

Use PEMDAS.

$3\times <!--\; \times \; -->4^3\times <!--\; \times \; -->7+24/6\times <!--\; \times \; -->7^2-<!--\; -\; -->9/3\times <!--\; \times \; -->8$

Use PEMDAS.

$8-<!--\; -\; -->\left(25-<!--\; -\; -->2^2\right)/7$

Use PEMDAS.

Evaluate the operations inside the parentheses first.

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

Use PEMDAS.

Evaluate the operations inside the parentheses first.

94 is not a perfect square.

The prime factorization of 94 is $2\times <!--\; \times \; -->47$. None of the factors is a pair.

Not rational

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

Two fractions, $\frac{a}{b}$ and $\frac{c}{d}$ are equivalent if $a\times <!--\; \times \; -->d=b\times <!--\; \times \; -->c$.

$a\times <!--\; \times \; -->d=8\times <!--\; \times \; -->26=208$

$b\times <!--\; \times \; -->c=14\times <!--\; \times \; -->12=168$

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 $\frac{\mathrm{◻<!--\; ◻\; -->}}{\mathrm{◻<!--\; ◻\; -->}}$.

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.)

$\mathrm{L}\mathrm{C}\mathrm{M}=2\times <!--\; \times \; -->2\times <!--\; \times \; -->3\times <!--\; \times \; -->3=36$

Rewrite the fractions with 36 as the denominator.

$\frac{4\times <!--\; \times \; -->4}{9\times <!--\; \times \; -->4}+\frac{7\times <!--\; \times \; -->3}{12\times <!--\; \times \; -->3}$

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

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.)

$\mathrm{L}\mathrm{C}\mathrm{M}=2\times <!--\; \times \; -->2\times <!--\; \times \; -->3\times <!--\; \times \; -->3\times <!--\; \times \; -->5\times <!--\; \times \; -->5\times <!--\; \times \; -->11=9,900$

Rewrite the fractions with 9,900 as the denominator.

$\frac{10\times <!--\; \times \; -->100}{99\times <!--\; \times \; -->100}-<!--\; -\; -->\frac{17\times <!--\; \times \; -->33}{300\times <!--\; \times \; -->33}$

$\frac{1,000}{9,900}-<!--\; -\; -->\frac{561}{9,900}$

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?

$26\times <!--\; \times \; -->3=78$

$95-<!--\; -\; -->78=17$

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.

$\frac{48}{30}=\frac{\cancel{3}\times <!--\; \times \; -->16}{\cancel{3}\times <!--\; \times \; -->10}$

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

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

${10}^5=100,000$

$\frac{17.03347\times <!--\; \times \; -->100,000}{1\times <!--\; \times \; -->100,000}=\frac{1,703,347}{100,000}$

$\frac{45}{88}\times <!--\; \times \; -->\frac{28}{75}$

Multiply the numerators and place that in the numerator.

Multiply the denominators and place that in the denominator.

$\frac{45\times <!--\; \times \; -->28}{88\times <!--\; \times \; -->75}$

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

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

$\frac{\cancel{60}\times <!--\; \times \; -->21}{\cancel{60}\times <!--\; \times \; -->110}$

$\frac{21}{110}$

$45.63÷<!--\; ÷\; -->17.13$ 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.

$\frac{46}{175}\times <!--\; \times \; -->\frac{285}{69}=\frac{13,110}{12,075}$

Simplify.

$\frac{38\times <!--\; \times \; -->\cancel{345}}{35\times <!--\; \times \; -->\cancel{345}}$

$\frac{38}{35}$

$\frac{3}{40}÷<!--\; ÷\; -->\frac{42}{55}$

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

$\frac{3}{40}\times <!--\; \times \; -->\frac{55}{42}=\frac{165}{1,680}$

Simplify.

$\frac{11\times <!--\; \times \; -->\cancel{15}}{112\times <!--\; \times \; -->\cancel{15}}$

$\frac{11}{112}$

${\left(\frac{3}{16}+\frac{7}{16}\right)}^2+\frac{1}{5}÷<!--\; ÷\; -->\frac{3}{10}$

Use PEMDAS.

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

Use PEMDAS.

Optional decimal form: $0.31\stackrel{¯<!--\; ¯\; -->}{90}$

The density property process:

Step 1: Add the two rational numbers.

$\frac{27}{13}+\frac{21}{10}=\frac{27\times <!--\; \times \; -->10}{13\times <!--\; \times \; -->10}+\frac{21\times <!--\; \times \; -->13}{10\times <!--\; \times \; -->13}$

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

$\frac{543}{130}$

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

$\frac{543}{130}\times <!--\; \times \; -->\frac{1}{2}=\frac{543}{260}$

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

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

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

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

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

30% of 2,400 calories

$\frac{30}{100}\times <!--\; \times \; -->2,400=720$

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 $\frac{n}{100}\times <!--\; \times \; -->x$.

$n$ 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 $\frac{n}{100}\times <!--\; \times \; -->1,000.$

$70=\frac{n}{100}\times <!--\; \times \; -->1,000$

$7=n$

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 $\frac{n}{100}\times <!--\; \times \; -->500.$

$425=\frac{n}{100}\times <!--\; \times \; -->500$

$85=n$

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 $\frac{n}{100}\times <!--\; \times \; -->500.$

$54=\frac{n}{100}\times <!--\; \times \; -->450$

$12=n$

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

40% of the total is $30.

$n\mathrm{\%<!--\; \%\; -->}$ of $x$ items is $\frac{n}{100}\times <!--\; \times \; -->x$.

The original price was $75.

This is a perfect square.

$\frac{12}{221}=\frac{3\times <!--\; \times \; -->4}{13\times <!--\; \times \; -->17}$

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: $\left(a\times <!--\; \times \; -->x\right)\pm <!--\; \pm \; -->\left(b\times <!--\; \times \; -->x\right)=\left(a\pm <!--\; \pm \; -->b\right)\times <!--\; \times \; -->x$.

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

$18\sqrt{15}$

$4.1\mathrm{\pi <!--\; \pi \; -->}+3.2\mathrm{\pi <!--\; \pi \; -->}$

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

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

$\left(4.1+3.2\right)\times <!--\; \times \; -->\mathrm{\pi <!--\; \pi \; -->}$ Add.

$7.3\mathrm{\pi <!--\; \pi \; -->}$

$84\sqrt{132}÷<!--\; ÷\; -->\left(14\sqrt{11}\right)$

Step 1: Divide the rational part.

$84÷<!--\; ÷\; -->14=6$

Step 2: If necessary, reduce the result of Step 1 to lowest terms.

Not necessary.

Step 3: Divide the irrational parts.

$\sqrt{132}÷<!--\; ÷\; -->\sqrt{11}=\sqrt{12}$

Step 4: If necessary, reduce the result from Step 3 to lowest terms.

$\sqrt{12}=\sqrt{4}\sqrt{3}=2\sqrt{3}$

Step 5: The result is the product of the rational and the irrational parts.

$6\left(2\sqrt{3}\right)=12\sqrt{3}$

$57\sqrt{792}÷<!--\; ÷\; -->\left(25\sqrt{2}\right)$

Step 1: Divide the rational part.

$57÷<!--\; ÷\; -->25=\frac{57}{25}$

Step 2: If necessary, reduce the result of Step 1 to lowest terms.

Not necessary.

Step 3: Divide the irrational parts.

$\sqrt{792}÷<!--\; ÷\; -->\sqrt{2}=\sqrt{\frac{792}{2}}=\sqrt{396}$

Step 4: If necessary, reduce the result from Step 3 to lowest terms.

$\sqrt{396}=\sqrt{36}\sqrt{11}=6\sqrt{11}$

Step 5: The result is the product of the rational and the irrational parts.

$\frac{57}{25}\left(6\sqrt{11}\right)=\frac{342}{25}\sqrt{11}$

$1.2\sqrt{21}\times <!--\; \times \; -->\left(4.5\sqrt{14}\right)$

Step 1: Multiply the rational part.

$1.2\times <!--\; \times \; -->4.5=5.4$

Step 2: If necessary, reduce the result of Step 1 to lowest terms.

Not necessary.

Step 3: Multiply the irrational parts.

$\sqrt{21}\times <!--\; \times \; -->\sqrt{14}=\sqrt{294}$

Step 4: If necessary, reduce the result from Step 3 to lowest terms.

$\sqrt{294}=\sqrt{49}\sqrt{6}=7\sqrt{6}$

Step 5: The result is the product of the rational and the irrational parts.

$5.4\left(7\sqrt{6}\right)=37.8\sqrt{6}$

$38\mathrm{\pi <!--\; \pi \; -->}÷<!--\; ÷\; -->\left(2\mathrm{\pi <!--\; \pi \; -->}\right)$

Step 1: Divide the rational part.

$38÷<!--\; ÷\; -->2=19$

Step 2: If necessary, reduce the result of Step 1 to lowest terms.

Not necessary.

Step 3: Divide the irrational parts.

$\mathrm{\pi <!--\; \pi \; -->}÷<!--\; ÷\; -->\mathrm{\pi <!--\; \pi \; -->}=1$

Step 4: If necessary, reduce the result from Step 3 to lowest terms.

Not necessary.

Step 5: The result is the product of the rational and the irrational parts.

$19\times <!--\; \times \; -->1=19$

$\frac{24}{\sqrt{15}}$

Multiply the numerator and denominator by $\sqrt{15}$.

$\frac{24\times <!--\; \times \; -->\sqrt{15}}{\sqrt{15}\times <!--\; \times \; -->\sqrt{15}}$

Simplify.

$\frac{24\sqrt{15}}{15}$

Divide out 3.

$\frac{8\sqrt{15}}{5}$

$\frac{11\sqrt{14}}{6\sqrt{21}}$

Multiply the numerator and denominator by $\sqrt{21}$.

$\frac{11\sqrt{14}\times <!--\; \times \; -->\sqrt{21}}{6\sqrt{21}\times <!--\; \times \; -->\sqrt{21}}$

Simplify.

$\frac{11\sqrt{14\times <!--\; \times \; -->21}}{6\times <!--\; \times \; -->21}$

Divide out 3.

$\frac{11\sqrt{49}\sqrt{6}}{6\times <!--\; \times \; -->21}$

$\frac{11\times <!--\; \times \; -->\cancel{7}\sqrt{6}}{6\times <!--\; \times \; -->\cancel{7}\times <!--\; \times \; -->3}$

$\frac{11\sqrt{6}}{18}$

$\frac{15}{5-<!--\; -\; -->\sqrt{13}}$

Step 1: Recognize that the denominator is the sum or difference of two numbers, one or both involving square roots. This means the conjugate can be used to remove the square root from the denominator.

Step 2: Multiply the numerator and denominator by the conjugate of the denominator: $5+\sqrt{13}$.

$\frac{15}{5-<!--\; -\; -->\sqrt{13}}\times <!--\; \times \; -->\frac{5+\sqrt{13}}{5+\sqrt{13}}$

Step 3: In the denominator, remember that a sum times a difference is a difference of squares.

$\frac{15}{5-<!--\; -\; -->\sqrt{13}}\times <!--\; \times \; -->\frac{5+\sqrt{13}}{5+\sqrt{13}}=\frac{15\times <!--\; \times \; -->\left(5+\sqrt{13}\right)}{5^2-<!--\; -\; -->{\left(\sqrt{13}\right)}^2}=\frac{15\times <!--\; \times \; -->\left(5+\sqrt{13}\right)}{25-<!--\; -\; -->13}=\frac{15\times <!--\; \times \; -->\left(5+\sqrt{13}\right)}{12}$

Step 4: In the numerator, apply the distributive property.

$\frac{15\times <!--\; \times \; -->\left(5+\sqrt{13}\right)}{12}=\frac{75+15\sqrt{13}}{12}$

Step 5: You can write the answer as two separate numbers by recalling that $\frac{a-<!--\; -\; -->b}{c}=\frac{a}{c}-<!--\; -\; -->\frac{b}{c}$.

$\frac{75+15\sqrt{13}}{12}=\frac{75}{12}+\frac{15\sqrt{13}}{12}$

Simplify.

$\frac{25}{4}+\frac{5\sqrt{13}}{4}$

Check Your Understanding

–4 belongs in the Z area as it is an integer.

13.863 belongs in the Q area as it is a rational number. Terminating decimals can be written as the ratio of two integers.

15 and 871 belong in the N area as they are natural numbers.

$-<!--\; -\; -->3\mathrm{\pi <!--\; \pi \; -->}$ and $5\sqrt{2}$ belong in the R area as they are real numbers. They are irrational and belong in none of the other areas.

When using the distributive property, multiplication distributes across addition.

$a\times <!--\; \times \; -->\left(b+c\right)=\left(a\times <!--\; \times \; -->b\right)+\left(a\times <!--\; \times \; -->c\right)$

The additive inverse property tells you that every number plus its negative results in zero.

$a+\left(-<!--\; -\; -->a\right)=0$

Determine the remainder when 93 is divided by 12.

93 ÷ 12 = 7 and a remainder. What is that remainder?

$12\times <!--\; \times \; -->7=84$

Subtract 84 from 93.

$93–84=9$

The remainder is 9.

The remainder is the solution to 93 mod 12.

93 mod 12 ≡ 9

Determine the remainder when 387 is divided by 12.

387 ÷ 12 = 32 and a remainder. What is that remainder?

$12\times <!--\; \times \; -->32=384$

Subtract 384 from 387.

$387–384=3$

The remainder is 3.

The remainder is the solution to 387 mod 12.

387 mod 12 ≡ 3

First, find 43 mod 12.

Determine the remainder when 43 is divided by 12.

43 ÷ 12 = 3 and a remainder. What is that remainder?

$12\times <!--\; \times \; -->3=36$

Subtract 36 from 43.

$43-<!--\; -\; -->36=7$

The remainder is 7.

The remainder is the solution to 43 mod 12.

43 mod 12 ≡ 7

Now, add 7 hours to 9:00.

The time 7 hours after 9:00 is 4:00.

First, find 34 mod 12.

Determine the remainder when 34 is divided by 12.

34 ÷ 12 = 2 and a remainder. What is that remainder?

$12\times <!--\; \times \; -->2=24$

Subtract 24 from 34.

$34–24=10$

The remainder is 10.

The remainder is the solution to 34 mod 12.

34 mod 12 ≡ 10

Now, subtract 10 hours from 7:00.

The time 10 hours before 7:00 was 9:00.

First, multiply 4 and 19 to get 76.

Now, find 76 mod 12.

Determine the remainder when 76 is divided by 12.

76 ÷ 12 = 6 and a remainder. What is that remainder?

$12\times <!--\; \times \; -->6=72$

Subtract 72 from 76.

$76–72=4$

The remainder is 4.

The remainder is the solution to 76 mod 12.

76 mod 12 ≡ 4

The product of 4 and 19 modulo 12 is 4.

You text every three hours for 15 times. You texted for 45 hours.

You need to find 45 modulo 12.

45 ÷ 12 = 3 and a remainder. What is that remainder?

$12\times <!--\; \times \; -->3=36$

Subtract 36 from 45.

$45–36=9$

The remainder is 9.

The remainder is the solution to 45 modulo 12.

45 modulo 12 ≡ 9

Nine hours after 8 AM is 5 PM.

If you prepared the meal 20 more times and you do it every 5 days, then 100 days have passed.

To find what day it is, you need to find 100 modulo 7.

100 ÷ 7 = 14 and a remainder. What is that remainder?

$7\times <!--\; \times \; -->14=98$

Subtract 98 from 100.

$100–98=2$

The remainder is 2.

The remainder is the solution to 100 modulo 7.

100 modulo 7 ≡ 2

Two days after Tuesday is Thursday.

It will be Thursday after you have prepared the meals 20 more times.

Use the product rule: $a^m\times <!--\; \times \; -->a^n=a^{m+n}$.

${12}^{13}\times <!--\; \times \; -->{12}^8={12}^{13+8}={12}^{21}$

Use the product rule: $a^m\times <!--\; \times \; -->a^n=a^{m+n}$.

$b^6\times <!--\; \times \; -->b^3=b^{6+3}=b^9$

Use the quotient rule: $\frac{a^n}{a^m}=a^{\left(n-<!--\; -\; -->m\right)}$.

$\frac{b^6}{b^4}=b^{\left(6-<!--\; -\; -->4\right)}=b^2$

Use the distributive rule for exponents: ${\left(a\times <!--\; \times \; -->b\right)}^n=a^n\times <!--\; \times \; -->b^n$.

${\left(2\times <!--\; \times \; -->19\right)}^{14}=2^{14}\times <!--\; \times \; -->{19}^{14}$

Use the distributive rule for exponents: ${\left(a\times <!--\; \times \; -->b\right)}^n=a^n\times <!--\; \times \; -->b^n$.

${\left(a\times <!--\; \times \; -->b\right)}^6=a^6\times <!--\; \times \; -->b^6$

Use the exponent distributive rule. When you have a fraction, $\frac{a}{b}$, raised to an exponent, n, then ${\left(\frac{a}{b}\right)}^n=\frac{a^n}{b^n}$.

${\left(\frac{14}{5}\right)}^9=\frac{{14}^9}{5^9}$

Use the exponent distributive rule. When you have a fraction, $\frac{a}{b}$, raised to an exponent, n, then ${\left(\frac{a}{b}\right)}^n=\frac{a^n}{b^n}$.

${\left(\frac{a}{18}\right)}^5=\frac{a^5}{{18}^5}$

Use the power rule. If you raise a non-zero base to an exponent n, and raise that to another exponent, m, you get the base raised to the product of the exponents, which is ${\left(a^n\right)}^m=a^{\left(n\times <!--\; \times \; -->m\right)}$.

${\left({11}^4\right)}^{12}={11}^{\left(4\times <!--\; \times \; -->12\right)}={11}^{48}$

Use the power rule. If you raise a non-zero base to an exponent n, and raise that to another exponent, m, you get the base raised to the product of the exponents, which is ${\left(a^n\right)}^m=a^{\left(n\times <!--\; \times \; -->m\right)}$.

${\left(a^7\right)}^6=a^{\left(7\times <!--\; \times \; -->6\right)}=a^{42}$

Use the negative exponent rule. $a^{-<!--\; -\; -->n}=\frac{1}{a^n}$ provided that a ≠ 0.

${12}^{-<!--\; -\; -->3}\times <!--\; \times \; -->7^5=\frac{1}{{12}^3}\times <!--\; \times \; -->7^5=\frac{7^5}{{12}^3}$

Use the negative exponent rule. $a^{-<!--\; -\; -->n}=\frac{1}{a^n}$ provided that a ≠ 0.

$c^{-<!--\; -\; -->7}\times <!--\; \times \; -->5^3=\frac{1}{c^7}\times <!--\; \times \; -->5^3=\frac{5^3}{c^7}$

Use the negative exponent rule. $a^{-<!--\; -\; -->n}=\frac{1}{a^n}$ provided that a ≠ 0.

The negative exponent rule lets you say $\frac{1}{{13}^8}$ is equal to ${13}^{-<!--\; -\; -->8}$.

$\frac{6^3}{{13}^8}=6^3\times <!--\; \times \; -->{13}^{-<!--\; -\; -->8}$

Use the negative exponent rule. $a^{-<!--\; -\; -->n}=\frac{1}{a^n}$ provided that a ≠ 0.

The negative exponent rule lets you say $\frac{1}{2^9}$ is equal to $2^{-<!--\; -\; -->9}$.

$\frac{c^5}{2^9}=c^5\times <!--\; \times \; -->2^{-<!--\; -\; -->9}$

${\left(\frac{7^9}{{10}^5\times <!--\; \times \; -->6^3}\right)}^8$

Use the exponent distributive rule. When you have a fraction, $\frac{a}{b}$, raised to an exponent, n, then ${\left(\frac{a}{b}\right)}^n=\frac{a^n}{b^n}$.

$\frac{{\left(7^9\right)}^8}{{\left({10}^5\times <!--\; \times \; -->6^3\right)}^8}$

Use the distributive rule for exponents in the denominator: ${\left(a\times <!--\; \times \; -->b\right)}^n=a^n\times <!--\; \times \; -->b^n$.

$\frac{{\left(7^9\right)}^8}{{\left({10}^5\right)}^8\times <!--\; \times \; -->{\left(6^3\right)}^8}$

Use the power rule. If you raise a non-zero base to an exponent n, and raise that to another exponent, m, you get the base raised to the product of the exponents.

${\left(a^n\right)}^m=a^{\left(n\times <!--\; \times \; -->m\right)}$

$\frac{7^{72}}{{10}^{40}\times <!--\; \times \; -->6^{24}}$

${\left(\frac{4}{a^9{b}^6}\right)}^2$

Use the exponent distributive rule. When you have a fraction, $\frac{a}{b}$, raised to an exponent, n, then ${\left(\frac{a}{b}\right)}^n=\frac{a^n}{b^n}$.

$\frac{{\left(4\right)}^2}{{\left(a^9{b}^6\right)}^2}$

Use the distributive rule for exponents in the denominator: ${\left(a\times <!--\; \times \; -->b\right)}^n=a^n\times <!--\; \times \; -->b^n$.

$\frac{{\left(4\right)}^2}{{\left(a^9\right)}^2\times <!--\; \times \; -->{\left(b^6\right)}^2}$

Use the power rule. If you raise a non-zero base to an exponent n, and raise that to another exponent, m, you get the base raised to the product of the exponents.

${\left(a^n\right)}^m=a^{\left(n\times <!--\; \times \; -->m\right)}$

$\frac{16}{a^{18}{b}^{12}}$

This is not a product of a number between 1 and 9 multiplied by 10 raised to an integer power.

42.67 is bigger than 9.

This is not a product of a number between 1 and 9 multiplied by 10 raised to an integer power.

$-<!--\; -\; -->80.91$ is bigger than 9.

–38300

Case 3: The absolute value of the number is 10 or larger.

Step 1: Count the number of digits that are to the left of the decimal point. Label this n.

$n=5$

Step 2: Write the digits of the number without the decimal place, if one was present. If the number was negative, include the sign.

–38300

Step 3: If there is more than one digit, place the decimal point after the first digit.

–3.8300

Step 4: Multiply the number from Step 3 by ${10}^{n-<!--\; -\; -->1}$.

$-<!--\; -\; -->3.83\times <!--\; \times \; -->{10}^4$