# 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?

what is the stm
is there industrial application of fullrenes. What is the method to prepare fullrene on large scale.?
Rafiq
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Damian
How we are making nano material?
what is a peer
What is meant by 'nano scale'?
What is STMs full form?
LITNING
scanning tunneling microscope
Sahil
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Santosh
Do u think that Graphene and Fullrene fiber can be used to make Air Plane body structure the lightest and strongest. Rafiq
Rafiq
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Mahi
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Rafiq
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write examples of Nano molecule?
Bob
The nanotechnology is as new science, to scale nanometric
brayan
nanotechnology is the study, desing, synthesis, manipulation and application of materials and functional systems through control of matter at nanoscale
Damian
Is there any normative that regulates the use of silver nanoparticles?
what king of growth are you checking .?
Renato
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why we need to study biomolecules, molecular biology in nanotechnology?
?
Kyle
yes I'm doing my masters in nanotechnology, we are being studying all these domains as well..
why?
what school?
Kyle
biomolecules are e building blocks of every organics and inorganic materials.
Joe
anyone know any internet site where one can find nanotechnology papers?
research.net
kanaga
sciencedirect big data base
Ernesto
Introduction about quantum dots in nanotechnology
what does nano mean?
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Bharti
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absolutely yes
Daniel
how to know photocatalytic properties of tio2 nanoparticles...what to do now
it is a goid question and i want to know the answer as well
Maciej
Abigail
for teaching engĺish at school how nano technology help us
Anassong
How can I make nanorobot?
Lily
Do somebody tell me a best nano engineering book for beginners?
there is no specific books for beginners but there is book called principle of nanotechnology
NANO
how can I make nanorobot?
Lily
what is fullerene does it is used to make bukky balls
are you nano engineer ?
s.
fullerene is a bucky ball aka Carbon 60 molecule. It was name by the architect Fuller. He design the geodesic dome. it resembles a soccer ball.
Tarell
what is the actual application of fullerenes nowadays?
Damian
That is a great question Damian. best way to answer that question is to Google it. there are hundreds of applications for buck minister fullerenes, from medical to aerospace. you can also find plenty of research papers that will give you great detail on the potential applications of fullerenes.
Tarell
how did you get the value of 2000N.What calculations are needed to arrive at it
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Good
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Shanjida
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s.
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
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