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

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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?

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.
LeBron needs 150 milliliters of a 30% solution of sulfuric acid for a lab experiment but only has access to a 25% and a 50% solution. How much of the 25% and how much of the 50% solution should he mix to make the 30% solution?
5%
Michael
hey everyone how to do algebra
Felecia answer 1.5 hours before he reaches her
I would like to solve the problem -6/2x
12x
Andrew
how
Christian
Does the x represent a number or does it need to be graphed ?
latonya
-3/x
Venugopal
-3x is correct
Atul
Arnold invested $64,000, some at 5.5% interest and the rest at 9%. How much did he invest at each rate if he received$4,500 in interest in one year?
Tickets for the community fair cost $12 for adults and$5 for children. On the first day of the fair, 312 tickets were sold for a total of $2204. How many adult tickets and how many child tickets were sold? Alpha Reply 220 gayla Three-fourths of the people at a concert are children. If there are 87 children, what is the total number of people at the concert? Tsimmuaj Reply Erica earned a total of$50,450 last year from her two jobs. The amount she earned from her job at the store was $1,250 more than four times the amount she earned from her job at the college. How much did she earn from her job at the college? Tsimmuaj Erica earned a total of$50,450 last year from her two jobs. The amount she earned from her job at the store was $1,250 more than four times the amount she earned from her job at the college. How much did she earn from her job at the college? Tsimmuaj ? Is there anything wrong with this passage I found the total sum for 2 jobs, but found why elaborate on extra If I total one week from the store *4 would = the month than the total is = x than x can't calculate 10 month of a year candido what would be wong candido 87 divided by 3 then multiply that by 4. 116 people total. Melissa the actual number that has 3 out of 4 of a whole pie candido was having a hard time finding Teddy use Matrices for the 2nd question Daniel One number is 11 less than the other number. If their sum is increased by 8, the result is 71. Find the numbers. Tsimmuaj Reply 26 + 37 = 63 + 8 = 71 gayla 26+37=63+8=71 ziad 11+52=63+8=71 Thisha how do we know the answer is correct? Thisha 23 is 11 less than 37. 23+37=63. 63+8=71. that is what the question asked for. gayla 23 +11 = 37. 23+37=63 63+8=71 Gayla by following the question. one number is 11 less than the other number 26+11=37 so 26+37=63+8=71 Gayla your answer did not fit the guidelines of the question 11 is 41 less than 52. gayla 71-8-11 =52 is this correct? Ruel let the number is 'x' and the other number is "x-11". if their sum is increased means: x+(x-11)+8 result will be 71. so x+(x-11)+8=71 2x-11+8=71 2x-3=71 2x=71+3 2x=74 1/2(2x=74)1/2 x=37 final answer tesfu just new Muwanga Amara currently sells televisions for company A at a salary of$17,000 plus a $100 commission for each television she sells. Company B offers her a position with a salary of$29,000 plus a $20 commission for each television she sells. How televisions would Amara need to sell for the options to be equal? Tsimmuaj Reply yes math Kenneth company A 13 company b 5. A 17,000+13×100=29,100 B 29,000+5×20=29,100 gayla need help with math to do tsi test Toocute me too Christian have you tried the TSI practice test ***tsipracticetest.com gayla DaMarcus and Fabian live 23 miles apart and play soccer at a park between their homes. DaMarcus rode his bike for 34 of an hour and Fabian rode his bike for 12 of an hour to get to the park. Fabian’s speed was 6 miles per hour faster than DaMarcus’s speed. Find the speed of both soccer players. gustavo Reply ? Ann DaMarcus: 16 mi/hr Fabian: 22 mi/hr Sherman Joy is preparing 20 liters of a 25% saline solution. She has only a 40% solution and a 10% solution in her lab. How many liters of the 40% solution and how many liters of the 10% solution should she mix to make the 25% solution? Wenda Reply 15 and 5 32 is 40% , & 8 is 10 % , & any 4 letters is 5%. Karen It felt that something is missing on the question like: 40% of what solution? 10% of what solution? Jhea its confusing Sparcast 3% & 2% to complete the 25% Sparcast because she already has 20 liters. Sparcast ok I was a little confused I agree 15% & 5% Sparcast 8,2 Karen Jim and Debbie earned$7200. Debbie earned \$1600 more than Jim earned. How much did they earned
5600
Gloria
1600
Gloria
Bebbie: 4,400 Jim: 2,800
Jhea
A river cruise boat sailed 80 miles down the Mississippi River for 4 hours. It took 5 hours to return. Find the rate of the cruise boat in still water and the rate of the current.
A veterinarian is enclosing a rectangular outdoor running area against his building for the dogs he cares for. He needs to maximize the area using 100 feet of fencing. The quadratic equation A=x(100−2x) gives the area, A , of the dog run for the length, x , of the building that will border the dog run. Find the length of the building that should border the dog run to give the maximum area, and then find the maximum area of the dog run.
ggfcc
Mike