10.2 Use multiplication properties of exponents  (Page 3/3)

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Simplify: ${\left(7n\right)}^{2}\left(2{n}^{12}\right).$

98 n 14

Simplify: ${\left(4m\right)}^{2}\left(3{m}^{3}\right).$

48 m 5

Simplify: ${\left(3{p}^{2}q\right)}^{4}{\left(2p{q}^{2}\right)}^{3}.$

Solution

 ${\left(3{p}^{2}q\right)}^{4}{\left(2p{q}^{2}\right)}^{3}$ Use the Power of a Product Property. ${3}^{4}{\left({p}^{2}\right)}^{4}{q}^{4}·{2}^{3}{p}^{3}{\left({q}^{2}\right)}^{3}$ Use the Power Property. $81{p}^{8}{q}^{4}·8{p}^{3}{q}^{6}$ Use the Commutative Property. $81·8·{p}^{8}·{p}^{3}·{q}^{4}·{q}^{6}$ Multiply the constants and add the exponents for each variable. $648{p}^{11}{q}^{10}$

Simplify: ${\left({u}^{3}{v}^{2}\right)}^{5}{\left(4u{v}^{4}\right)}^{3}.$

64 u 18 v 22

Simplify: ${\left(5{x}^{2}{y}^{3}\right)}^{2}{\left(3x{y}^{4}\right)}^{3}.$

675 x 7 y 18

Multiply monomials

Since a monomial    is an algebraic expression, we can use the properties for simplifying expressions with exponents to multiply the monomials.

Multiply: $\left(4{x}^{2}\right)\left(-5{x}^{3}\right).$

Solution

 $\left(4{x}^{2}\right)\left(-5{x}^{3}\right)$ Use the Commutative Property to rearrange the factors. $4·\left(-5\right)·{x}^{2}·{x}^{3}$ Multiply. $-20{x}^{5}$

Multiply: $\left(7{x}^{7}\right)\left(-8{x}^{4}\right).$

−56 x 11

Multiply: $\left(-9{y}^{4}\right)\left(-6{y}^{5}\right).$

54 y 9

Multiply: $\left(\frac{3}{4}\phantom{\rule{0.1em}{0ex}}{c}^{3}d\right)\left(12c{d}^{2}\right).$

Solution

 $\left(\frac{3}{4}\phantom{\rule{0.1em}{0ex}}{c}^{3}d\right)\left(12c{d}^{2}\right)$ Use the Commutative Property to rearrange the factors. $\frac{3}{4}·12·{c}^{3}·c·d·{d}^{2}$ Multiply. $9{c}^{4}{d}^{3}$

Multiply: $\left(\frac{4}{5}\phantom{\rule{0.1em}{0ex}}{m}^{4}{n}^{3}\right)\left(15m{n}^{3}\right).$

12 m 5 n 6

Multiply: $\left(\frac{2}{3}\phantom{\rule{0.1em}{0ex}}{p}^{5}q\right)\left(18{p}^{6}{q}^{7}\right).$

12 p 11 q 8

Key concepts

• Exponential Notation

This is read $a$ to the ${m}^{\mathrm{th}}$ power.

• Product Property of Exponents
• If $a$ is a real number and $m,n$ are counting numbers, then
${a}^{m}·{a}^{n}={a}^{m+n}$
• To multiply with like bases, add the exponents.
• Power Property for Exponents
• If $a$ is a real number and $m,n$ are counting numbers, then
${\left({a}^{m}\right)}^{n}={a}^{m\cdot n}$
• Product to a Power Property for Exponents
• If $a$ and $b$ are real numbers and $m$ is a whole number, then
${\left(ab\right)}^{m}={a}^{m}{b}^{m}$

Practice makes perfect

Simplify Expressions with Exponents

In the following exercises, simplify each expression with exponents.

${4}^{5}$

1,024

${10}^{3}$

${\left(\frac{1}{2}\right)}^{2}$

$\frac{1}{4}$

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

${\left(0.2\right)}^{3}$

0.008

${\left(0.4\right)}^{3}$

${\left(-5\right)}^{4}$

625

${\left(-3\right)}^{5}$

${-5}^{4}$

−625

${-3}^{5}$

${-10}^{4}$

−10,000

${-2}^{6}$

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

$-\frac{8}{27}$

${\left(-\frac{1}{4}\right)}^{4}$

$-{0.5}^{2}$

−.25

$-{0.1}^{4}$

Simplify Expressions Using the Product Property of Exponents

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

${x}^{3}·{x}^{6}$

x 9

${m}^{4}·{m}^{2}$

$a·{a}^{4}$

a 5

${y}^{12}·y$

${3}^{5}·{3}^{9}$

3 14

${5}^{10}·{5}^{6}$

$z·{z}^{2}·{z}^{3}$

z 6

$a·{a}^{3}·{a}^{5}$

${x}^{a}·{x}^{2}$

x a +2

${y}^{p}·{y}^{3}$

${y}^{a}·{y}^{b}$

y a + b

${x}^{p}·{x}^{q}$

Simplify Expressions Using the Power Property of Exponents

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

${\left({u}^{4}\right)}^{2}$

u 8

${\left({x}^{2}\right)}^{7}$

${\left({y}^{5}\right)}^{4}$

y 20

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

${\left({10}^{2}\right)}^{6}$

10 12

${\left({2}^{8}\right)}^{3}$

${\left({x}^{15}\right)}^{6}$

x 90

${\left({y}^{12}\right)}^{8}$

${\left({x}^{2}\right)}^{y}$

x 2 y

${\left({y}^{3}\right)}^{x}$

${\left({5}^{x}\right)}^{y}$

5 x y

${\left({7}^{a}\right)}^{b}$

Simplify Expressions Using the Product to a Power Property

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

${\left(5a\right)}^{2}$

25 a 2

${\left(7x\right)}^{2}$

${\left(-6m\right)}^{3}$

−216 m 3

${\left(-9n\right)}^{3}$

${\left(4rs\right)}^{2}$

16 r 2 s 2

${\left(5ab\right)}^{3}$

${\left(4xyz\right)}^{4}$

256 x 4 y 4 z 4

${\left(-5abc\right)}^{3}$

Simplify Expressions by Applying Several Properties

In the following exercises, simplify each expression.

${\left({x}^{2}\right)}^{4}·{\left({x}^{3}\right)}^{2}$

x 14

${\left({y}^{4}\right)}^{3}·{\left({y}^{5}\right)}^{2}$

${\left({a}^{2}\right)}^{6}·{\left({a}^{3}\right)}^{8}$

a 36

${\left({b}^{7}\right)}^{5}·{\left({b}^{2}\right)}^{6}$

${\left(3x\right)}^{2}\left(5x\right)$

45 x 3

${\left(2y\right)}^{3}\left(6y\right)$

${\left(5a\right)}^{2}{\left(2a\right)}^{3}$

200 a 5

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

${\left(2{m}^{6}\right)}^{3}$

8 m 18

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

${\left(10{x}^{2}y\right)}^{3}$

1,000 x 6 y 3

${\left(2m{n}^{4}\right)}^{5}$

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

16 a 12 b 8

${\left(-10{u}^{2}{v}^{4}\right)}^{3}$

${\left(\frac{2}{3}\phantom{\rule{0.1em}{0ex}}{x}^{2}y\right)}^{3}$

$\frac{8}{27}\phantom{\rule{0.1em}{0ex}}{x}^{6}{y}^{3}$

${\left(\frac{7}{9}\phantom{\rule{0.1em}{0ex}}p{q}^{4}\right)}^{2}$

${\left(8{a}^{3}\right)}^{2}{\left(2a\right)}^{4}$

1,024 a 10

${\left(5{r}^{2}\right)}^{3}{\left(3r\right)}^{2}$

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

25,000 p 24

${\left(4{x}^{3}\right)}^{3}{\left(2{x}^{5}\right)}^{4}$

${\left(\frac{1}{2}\phantom{\rule{0.1em}{0ex}}{x}^{2}{y}^{3}\right)}^{4}{\left(4{x}^{5}{y}^{3}\right)}^{2}$

x 18 y 18

${\left(\frac{1}{3}\phantom{\rule{0.1em}{0ex}}{m}^{3}{n}^{2}\right)}^{4}{\left(9{m}^{8}{n}^{3}\right)}^{2}$

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

144 m 8 n 22

${\left(2p{q}^{4}\right)}^{3}{\left(5{p}^{6}q\right)}^{2}$

Multiply Monomials

In the following exercises, multiply the following monomials.

$\left(12{x}^{2}\right)\left(-5{x}^{4}\right)$

−60 x 6

$\left(-10{y}^{3}\right)\left(7{y}^{2}\right)$

$\left(-8{u}^{6}\right)\left(-9u\right)$

72 u 7

$\left(-6{c}^{4}\right)\left(-12c\right)$

$\left(\frac{1}{5}\phantom{\rule{0.1em}{0ex}}{r}^{8}\right)\left(20{r}^{3}\right)$

4 r 11

$\left(\frac{1}{4}\phantom{\rule{0.1em}{0ex}}{a}^{5}\right)\left(36{a}^{2}\right)$

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

36 a 5 b 7

$\left(6{m}^{4}{n}^{3}\right)\left(7m{n}^{5}\right)$

$\left(\frac{4}{7}\phantom{\rule{0.1em}{0ex}}x{y}^{2}\right)\left(14x{y}^{3}\right)$

8 x 2 y 5

$\left(\frac{5}{8}\phantom{\rule{0.1em}{0ex}}{u}^{3}v\right)\left(24{u}^{5}v\right)$

$\left(\frac{2}{3}\phantom{\rule{0.1em}{0ex}}{x}^{2}y\right)\left(\frac{3}{4}\phantom{\rule{0.1em}{0ex}}x{y}^{2}\right)$

$\frac{1}{2}\phantom{\rule{0.1em}{0ex}}{x}^{3}{y}^{3}$

$\left(\frac{3}{5}\phantom{\rule{0.1em}{0ex}}{m}^{3}{n}^{2}\right)\left(\frac{5}{9}\phantom{\rule{0.1em}{0ex}}{m}^{2}{n}^{3}\right)$

Everyday math

Email Janet emails a joke to six of her friends and tells them to forward it to six of their friends, who forward it to six of their friends, and so on. The number of people who receive the email on the second round is ${6}^{2},$ on the third round is ${6}^{3},$ as shown in the table. How many people will receive the email on the eighth round? Simplify the expression to show the number of people who receive the email.

Round Number of people
$1$ $6$
$2$ ${6}^{2}$
$3$ ${6}^{3}$
$\dots$ $\dots$
$8$ $?$

1,679,616

Salary Raul’s boss gives him a $\text{5%}$ raise every year on his birthday. This means that each year, Raul’s salary is $1.05$ times his last year’s salary. If his original salary was $\text{40,000}$ , his salary after $1$ year was $\text{40,000}\left(1.05\right),$ after $2$ years was $\text{40,000}{\left(1.05\right)}^{2},$ after $3$ years was $\text{40,000}{\left(1.05\right)}^{3},$ as shown in the table below. What will Raul’s salary be after $10$ years? Simplify the expression, to show Raul’s salary in dollars.

Year Salary
$1$ $\text{40,000}\left(1.05\right)$
$2$ $\text{40,000}{\left(1.05\right)}^{2}$
$3$ $\text{40,000}{\left(1.05\right)}^{3}$
$\dots$ $\dots$
$10$ $?$

Writing exercises

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

Explain why ${-5}^{3}={\left(-5\right)}^{3}$ but ${-5}^{4}\ne {\left(-5\right)}^{4}.$

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

Explain why ${x}^{3}·{x}^{5}$ is ${x}^{8},$ and not ${x}^{15}.$

Self check

After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

After reviewing this checklist, what will you do to become confident for all objectives?

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