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

 Page 3 / 3

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 $x·x={x}^{2}.$

Answers will vary.

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?

Answers will vary.

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?

#### Questions & Answers

explain and give four Example hyperbolic function
Lukman Reply
The denominator of a certain fraction is 9 more than the numerator. If 6 is added to both terms of the fraction, the value of the fraction becomes 2/3. Find the original fraction. 2. The sum of the least and greatest of 3 consecutive integers is 60. What are the valu
SABAL Reply
1. x + 6 2 -------------- = _ x + 9 + 6 3 x + 6 3 ----------- x -- (cross multiply) x + 15 2 3(x + 6) = 2(x + 15) 3x + 18 = 2x + 30 (-2x from both) x + 18 = 30 (-18 from both) x = 12 Test: 12 + 6 18 2 -------------- = --- = --- 12 + 9 + 6 27 3
Pawel
2. (x) + (x + 2) = 60 2x + 2 = 60 2x = 58 x = 29 29, 30, & 31
Pawel
ok
Ifeanyi
on number 2 question How did you got 2x +2
Ifeanyi
combine like terms. x + x + 2 is same as 2x + 2
Pawel
Mark and Don are planning to sell each of their marble collections at a garage sale. If Don has 1 more than 3 times the number of marbles Mark has, how many does each boy have to sell if the total number of marbles is 113?
mariel Reply
Mark = x,. Don = 3x + 1 x + 3x + 1 = 113 4x = 112, x = 28 Mark = 28, Don = 85, 28 + 85 = 113
Pawel
how do I set up the problem?
Harshika Reply
what is a solution set?
Harshika
find the subring of gaussian integers?
Rofiqul
hello, I am happy to help!
Shirley Reply
please can go further on polynomials quadratic
Abdullahi
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Mark
I need quadratic equation link to Alpa Beta
Abdullahi Reply
find the value of 2x=32
Felix Reply
divide by 2 on each side of the equal sign to solve for x
corri
X=16
Michael
Want to review on complex number 1.What are complex number 2.How to solve complex number problems.
Beyan
yes i wantt to review
Mark
use the y -intercept and slope to sketch the graph of the equation y=6x
Only Reply
how do we prove the quadratic formular
Seidu Reply
please help me prove quadratic formula
Darius
hello, if you have a question about Algebra 2. I may be able to help. I am an Algebra 2 Teacher
Shirley Reply
thank you help me with how to prove the quadratic equation
Seidu
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Opoku
what is math number
Tric Reply
4
Trista
x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
Sidiki Reply
can you teacch how to solve that🙏
Mark
Solve for the first variable in one of the equations, then substitute the result into the other equation. Point For: (6111,4111,−411)(6111,4111,-411) Equation Form: x=6111,y=4111,z=−411x=6111,y=4111,z=-411
Brenna
(61/11,41/11,−4/11)
Brenna
x=61/11 y=41/11 z=−4/11 x=61/11 y=41/11 z=-4/11
Brenna
Need help solving this problem (2/7)^-2
Simone Reply
x+2y-z=7
Sidiki
what is the coefficient of -4×
Mehri Reply
-1
Shedrak
the operation * is x * y =x + y/ 1+(x × y) show if the operation is commutative if x × y is not equal to -1
Alfred Reply
A soccer field is a rectangle 130 meters wide and 110 meters long. The coach asks players to run from one corner to the other corner diagonally across. What is that distance, to the nearest tenths place.
Kimberly Reply
Jeannette has $5 and$10 bills in her wallet. The number of fives is three more than six times the number of tens. Let t represent the number of tens. Write an expression for the number of fives.
August Reply
What is the expressiin for seven less than four times the number of nickels
Leonardo Reply
How do i figure this problem out.
how do you translate this in Algebraic Expressions
linda Reply
why surface tension is zero at critical temperature
Shanjida
I think if critical temperature denote high temperature then a liquid stats boils that time the water stats to evaporate so some moles of h2o to up and due to high temp the bonding break they have low density so it can be a reason
s.
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
Crystal Reply
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?
Chris Reply

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