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

 Page 3 / 3

Simplify: ${\left(5n\right)}^{2}\left(3{n}^{10}\right)$ ${\left({c}^{4}{d}^{2}\right)}^{5}{\left(3c{d}^{5}\right)}^{4}.$

$75{n}^{12}$ $81{c}^{24}{d}^{30}$

Simplify: ${\left({a}^{3}{b}^{2}\right)}^{6}{\left(4a{b}^{3}\right)}^{4}$ ${\left(2x\right)}^{3}\left(5{x}^{7}\right).$

$256{a}^{22}{b}^{24}$ $40{x}^{10}$

## Multiply monomials

Since a monomial    is an algebraic expression, we can use the properties of exponents to multiply monomials.

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

## Solution

$\begin{array}{cccc}& & & \hfill \left(3{x}^{2}\right)\left(-4{x}^{3}\right)\hfill \\ \text{Use the Commutative Property to rearrange the terms.}\hfill & & & \hfill 3·\left(-4\right)·{x}^{2}·{x}^{3}\hfill \\ \text{Multiply.}\hfill & & & \hfill -12{x}^{5}\hfill \end{array}$

Multiply: $\left(5{y}^{7}\right)\left(-7{y}^{4}\right).$

$-35{y}^{11}$

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

$54{b}^{9}$

Multiply: $\left(\frac{5}{6}{x}^{3}y\right)\left(12x{y}^{2}\right).$

## Solution

$\begin{array}{cccc}& & & \hfill \left(\frac{5}{6}{x}^{3}y\right)\left(12x{y}^{2}\right)\hfill \\ \text{Use the Commutative Property to rearrange the terms.}\hfill & & & \hfill \frac{5}{6}·12·{x}^{3}·x·y·{y}^{2}\hfill \\ \text{Multiply.}\hfill & & & \hfill 10{x}^{4}{y}^{3}\hfill \end{array}$

Multiply: $\left(\frac{2}{5}{a}^{4}{b}^{3}\right)\left(15a{b}^{3}\right).$

$6{a}^{5}{b}^{6}$

Multiply: $\left(\frac{2}{3}{r}^{5}s\right)\left(12{r}^{6}{s}^{7}\right).$

$8{r}^{11}{s}^{8}$

Access these online resources for additional instruction and practice with using multiplication properties of exponents:

## Key concepts

• Exponential Notation
• Properties of Exponents
• If $a,b$ are real numbers and $m,n$ are whole numbers, then
$\begin{array}{cccccc}\mathbf{\text{Product Property}}\hfill & & & \hfill {a}^{m}·{a}^{n}& =\hfill & {a}^{m+n}\hfill \\ \mathbf{\text{Power Property}}\hfill & & & \hfill {\left({a}^{m}\right)}^{n}& =\hfill & {a}^{m·n}\hfill \\ \mathbf{\text{Product to a Power}}\hfill & & & \hfill {\left(ab\right)}^{m}& =\hfill & {a}^{m}{b}^{m}\hfill \end{array}$

## Practice makes perfect

Simplify Expressions with Exponents

In the following exercises, simplify each expression with exponents.

${3}^{5}$
${9}^{1}$
${\left(\frac{1}{3}\right)}^{2}$
${\left(0.2\right)}^{4}$

${10}^{4}$
${17}^{1}$
${\left(\frac{2}{9}\right)}^{2}$
${\left(0.5\right)}^{3}$

10,000 17 $\frac{4}{81}$ 0.125

${2}^{6}$
${14}^{1}$
${\left(\frac{2}{5}\right)}^{3}$
${\left(0.7\right)}^{2}$

${8}^{3}$
${8}^{1}$
${\left(\frac{3}{4}\right)}^{3}$
${\left(0.4\right)}^{3}$

512 8 $\frac{27}{64}$
0.064

${\left(-6\right)}^{4}$
$\text{−}{6}^{4}$

${\left(-2\right)}^{6}$
$\text{−}{2}^{6}$

64 $-64$

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

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

$-\frac{4}{9}$
$\frac{4}{9}$

$\text{−}{0.5}^{2}$
${\left(-0.5\right)}^{2}$

$\text{−}{0.1}^{4}$
${\left(-0.1\right)}^{4}$

$-0.001$ 0.001

Simplify Expressions Using the Product Property for Exponents

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

${d}^{3}·{d}^{6}$

${x}^{4}·{x}^{2}$

${x}^{6}$

${n}^{19}·{n}^{12}$

${q}^{27}·{q}^{15}$

${q}^{42}$

${4}^{5}·{4}^{9}$ ${8}^{9}·8$

${3}^{10}·{3}^{6}$ $5·{5}^{4}$

${3}^{16}$ ${5}^{5}$

$y·{y}^{3}$ ${z}^{25}·{z}^{8}$

${w}^{5}·{w}^{}$ ${u}^{41}·{u}^{53}$

${w}^{6}$ ${u}^{94}$

$w·{w}^{2}·{w}^{3}$

$y·{y}^{3}·{y}^{5}$

${y}^{9}$

${a}^{4}·{a}^{3}·{a}^{9}$

${c}^{5}·{c}^{11}·{c}^{2}$

${c}^{18}$

${m}^{x}·{m}^{3}$

${n}^{y}·{n}^{2}$

${n}^{y+2}$

${y}^{a}·{y}^{b}$

${x}^{p}·{x}^{q}$

${x}^{p+q}$

Simplify Expressions Using the Power Property for Exponents

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

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

${\left({b}^{2}\right)}^{7}$ ${\left({3}^{8}\right)}^{2}$

${b}^{14}$ ${3}^{16}$

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

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

${x}^{2y}$ ${7}^{ab}$

Simplify Expressions Using the Product to a Power Property

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

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

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

$25{x}^{2}$ $16{a}^{2}{b}^{2}$

${\left(-4m\right)}^{3}$ ${\left(5ab\right)}^{3}$

${\left(-7n\right)}^{3}$ ${\left(3xyz\right)}^{4}$

$-343{n}^{3}$ $81{x}^{4}{y}^{4}{z}^{4}$

Simplify Expressions by Applying Several Properties

In the following exercises, simplify each expression.

${\left({y}^{2}\right)}^{4}·{\left({y}^{3}\right)}^{2}$
${\left(10{a}^{2}b\right)}^{3}$

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

${w}^{22}$ $32{x}^{5}{y}^{20}$

${\left(-2{r}^{3}{s}^{2}\right)}^{4}$
${\left({m}^{5}\right)}^{3}·{\left({m}^{9}\right)}^{4}$

${\left(-10{q}^{2}{p}^{4}\right)}^{3}$
${\left({n}^{3}\right)}^{10}·{\left({n}^{5}\right)}^{2}$

$-1000{q}^{6}{p}^{12}$ ${n}^{40}$

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

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

$48{y}^{4}$ $25,000{k}^{24}$

${\left(5a\right)}^{2}{\left(2a\right)}^{3}$
${\left(\frac{1}{2}{y}^{2}\right)}^{3}{\left(\frac{2}{3}y\right)}^{2}$

${\left(4b\right)}^{2}{\left(3b\right)}^{3}$
${\left(\frac{1}{2}{j}^{2}\right)}^{5}{\left(\frac{2}{5}{j}^{3}\right)}^{2}$

$432{b}^{5}$ $\frac{1}{200}{j}^{16}$

${\left(\frac{2}{5}{x}^{2}y\right)}^{3}$
${\left(\frac{8}{9}x{y}^{4}\right)}^{2}$

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

$128{r}^{8}$ $\frac{1}{200}{j}^{16}$

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

Multiply Monomials

In the following exercises, multiply the monomials.

$\left(6{y}^{7}\right)\left(-3{y}^{4}\right)$

$-18{y}^{11}$

$\left(-10{x}^{5}\right)\left(-3{x}^{3}\right)$

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

$72{u}^{7}$

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

$\left(\frac{1}{5}{f}^{8}\right)\left(20{f}^{3}\right)$

$4{f}^{11}$

$\left(\frac{1}{4}{d}^{5}\right)\left(36{d}^{2}\right)$

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

$36{a}^{5}{b}^{7}$

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

$\left(\frac{4}{7}r{s}^{2}\right)\left(14r{s}^{3}\right)$

$8{r}^{2}{s}^{5}$

$\left(\frac{5}{8}{x}^{3}y\right)\left(24{x}^{5}y\right)$

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

$\frac{1}{2}{x}^{3}{y}^{3}$

$\left(\frac{3}{5}{m}^{3}{n}^{2}\right)\left(\frac{5}{9}{m}^{2}{n}^{3}\right)$

Mixed Practice

In the following exercises, simplify each expression.

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

${x}^{14}$

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

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

${a}^{36}$

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

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

$8{m}^{18}$

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

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

$1000{x}^{6}{y}^{3}$

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

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

$16{a}^{12}{b}^{8}$

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

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

$\frac{8}{27}{x}^{6}{y}^{3}$

${\left(\frac{7}{9}p{q}^{4}\right)}^{2}$

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

$1024{a}^{10}$

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

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

$25000{p}^{24}$

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

${\left(\frac{1}{2}{x}^{2}{y}^{3}\right)}^{4}{\left(4{x}^{5}{y}^{3}\right)}^{2}$

${x}^{18}{y}^{18}$

${\left(\frac{1}{3}{m}^{3}{n}^{2}\right)}^{4}{\left(9{m}^{8}{n}^{3}\right)}^{2}$

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

$144{m}^{8}{n}^{22}$

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

## Everyday math

Email Kate emails a flyer to ten of her friends and tells them to forward it to ten of their friends, who forward it to ten of their friends, and so on. The number of people who receive the email on the second round is ${10}^{2}$ , on the third round is ${10}^{3}$ , as shown in the table below. How many people will receive the email on the sixth round? Simplify the expression to show the number of people who receive the email.

Round Number of people
1 10
2 ${10}^{2}$
3 ${10}^{3}$
6 ?

$1,000,000$

Salary Jamal’s boss gives him a 3% raise every year on his birthday. This means that each year, Jamal’s salary is 1.03 times his last year’s salary. If his original salary was $35,000, his salary after 1 year was $\text{}35,000\left(1.03\right)$ , after 2 years was $\text{}35,000{\left(1.03\right)}^{2}$ , after 3 years was $\text{}35,000{\left(1.03\right)}^{3}$ , as shown in the table below. What will Jamal’s salary be after 10 years? Simplify the expression, to show Jamal’s salary in dollars. Year Salary 1 $\text{}35,000\left(1.03\right)$ 2 $\text{}35,000{\left(1.03\right)}^{2}$ 3 $\text{}35,000{\left(1.03\right)}^{3}$ 10 ? Clearance A department store is clearing out merchandise in order to make room for new inventory. The plan is to mark down items by 30% each week. This means that each week the cost of an item is 70% of the previous week’s cost. If the original cost of a sofa was$1,000, the cost for the first week would be $\text{}1,000\left(0.70\right)$ and the cost of the item during the second week would be $\text{}1,000{\left(0.70\right)}^{2}$ . Complete the table shown below. What will be the cost of the sofa during the fifth week? Simplify the expression, to show the cost in dollars.

Week Cost
1 $\text{}1,000\left(0.70\right)$
2 $\text{}1,000{\left(0.70\right)}^{2}$
3
8 ?

$168.07 Depreciation Once a new car is driven away from the dealer, it begins to lose value. Each year, a car loses 10% of its value. This means that each year the value of a car is 90% of the previous year’s value. If a new car was purchased for$20,000, the value at the end of the first year would be $\text{}20,000\left(0.90\right)$ and the value of the car after the end of the second year would be $\text{}20,000{\left(0.90\right)}^{2}$ . Complete the table shown below. What will be the value of the car at the end of the eighth year? Simplify the expression, to show the value in dollars.

Week Cost
1 $\text{}20,000\left(0.90\right)$
2 $\text{}20,000{\left(0.90\right)}^{2}$
3
4
5 ?

## Writing exercises

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

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

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 goals?

hello, I have algebra phobia. Subtracting negative numbers always seem to get me confused.
what do you need help in?
Felix
Heather
look at the numbers if they have different signs, it's like subtracting....but you keep the sign of the largest number...
Felix
for example.... -19 + 7.... different signs...subtract.... 12 keep the sign of the "largest" number 19 is bigger than 7.... 19 has the negative sign... Therefore, -12 is your answer...
Felix
—12
Thanks Felix.l also get confused with signs.
Esther
Thank you for this
Shatey
ty
Graham
Bruce drives his car for his job. The equation R=0.575m+42 models the relation between the amount in dollars, R, that he is reimbursed and the number of miles, m, he drives in one day. Find the amount Bruce is reimbursed on a day when he drives 220 miles
168.50=R
Heather
john is 5years older than wanjiru.the sum of their years is27years.what is the age of each
46
mustee
j 17 w 11
Joseph
john is 16. wanjiru is 11.
Felix
27-5=22 22÷2=11 11+5=16
Joyce
I don't see where the answers are.
Ed
Cindy and Richard leave their dorm in Charleston at the same time. Cindy rides her bicycle north at a speed of 18 miles per hour. Richard rides his bicycle south at a speed of 14 miles per hour. How long will it take them to be 96 miles apart?
3
Christopher
18t+14t=96 32t=96 32/96 3
Christopher
show that a^n-b^2n is divisible by a-b
What does 3 times your weight right now
Use algebra to combine 39×5 and the half sum of travel of 59+30
Cherokee
What is the segment of 13? Explain
Cherokee
my weight is 49. So 3 times is 147
Cherokee
kg to lbs you goin to convert 2.2 or one if the same unit your going to time your body weight by 3. example if my body weight is 210lb. what would be my weight if I was 3 times as much in kg. that's you do 210 x3 = 630lb. then 630 x 2.2= .... hope this helps
tyler
How to convert grams to pounds?
paul
What is the lcm of 340
Yes
Cherokee
How many numbers each equal to y must be taken to make 15xy
15x
Martin
15x
Asamoah
15x
Hugo
1y
Tom
1y x 15y
Tom
find the equation whose roots are 1 and 2
(x - 2)(x -1)=0 so equation is x^2-x+2=0
Ranu
I believe it's x^2-3x+2
NerdNamedGerg
because the X's multiply by the -2 and the -1 and than combine like terms
NerdNamedGerg
find the equation whose roots are -1 and 4
Ans = ×^2-3×+2
Gee
find the equation whose roots are -2 and -1
(×+1)(×-4) = x^2-3×-4
Gee
yeah
Asamoah
there's a chatting option in the app wow
Nana
That's cool cool
Nana
Nice to meet you all
Nana
you too.
Joan
😃
Nana
Hey you all there are several Free Apps that can really help you to better solve type Equations.
Debra
Debra, which apps specifically. ..?
Nana
am having a course in elementary algebra ,any recommendations ?
samuel
Samuel Addai, me too at ucc elementary algebra as part of my core subjects in science
Nana
me too as part of my core subjects in R M E
Ken
at ABETIFI COLLEGE OF EDUCATION
Ken
ok great. Good to know.
Joan
5x + 1/3= 2x + 1/2
sanam
Plz solve this
sanam
5x - 3x = 1/2 - 1/3 2x = 1/6 x = 1/12
Ranu
Thks ranu
sanam
Erica
the previous equation should be 3x = 1/6 x=1/18
Sriram
for the new one 10x + 2x = 38 - 14
Sriram
12x = 24 x=2
Sriram
10x + 14 = -2x +38 10x + 2x = 38 - 14 12x = 24 divide both sides by the coefficient of x, which is 12 therefore × = 2
vida
a trader gains 20 rupees loses 42 rupees and then gains ten rupees Express algebraically the result of his transactions
a trader gains 20 rupees loses 42 rupees and then gains 10 rupees Express algebraically the result of his three transactions
vinaya
a trader gains 20 rupees loses 42 rupees and then gains 10 rupees Express algebraically the result of his three transactions
vinaya
a trader gains 20 rupees loses 42 rupees and then gains 10 rupees Express algebraically the result of his three transactions
vinaya
Kim is making eight gallons of punch from fruit juice and soda. The fruit juice costs $6.04 per gallon and the soda costs$4.28 per gallon. How much fruit juice and how much soda should she use so that the punch costs \$5.71 per gallon?
(a+b)(p+q+r)(b+c)(p+q+r)(c+a) (p+q+r)
really
Asamoah
4x-7y=8 2x-7y=1 what is the answer?
x=7/2 & y=6/7
Pbp
x=7/2 & y=6/7 use Elimination
Debra
true
bismark
factoriz e
usman
4x-7y=8 X=7/4y+2 and 2x-7y=1 x=7/2y+1/2
Peggie
Frank
thanks
Ramil
x=7/2 y=6/7
Asamoah
Thanks , all of you are correct.
Joseph