# 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 $x·x={x}^{2}.$

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?

#### Questions & Answers

can someone help me with some logarithmic and exponential equations.
sure. what is your question?
20/(×-6^2)
Salomon
okay, so you have 6 raised to the power of 2. what is that part of your answer
I don't understand what the A with approx sign and the boxed x mean
it think it's written 20/(X-6)^2 so it's 20 divided by X-6 squared
Salomon
I'm not sure why it wrote it the other way
Salomon
I got X =-6
Salomon
ok. so take the square root of both sides, now you have plus or minus the square root of 20= x-6
oops. ignore that.
so you not have an equal sign anywhere in the original equation?
Commplementary angles
hello
Sherica
im all ears I need to learn
Sherica
right! what he said ⤴⤴⤴
Tamia
what is a good calculator for all algebra; would a Casio fx 260 work with all algebra equations? please name the cheapest, thanks.
a perfect square v²+2v+_
kkk nice
algebra 2 Inequalities:If equation 2 = 0 it is an open set?
or infinite solutions?
Kim
The answer is neither. The function, 2 = 0 cannot exist. Hence, the function is undefined.
Al
y=10×
if |A| not equal to 0 and order of A is n prove that adj (adj A = |A|
rolling four fair dice and getting an even number an all four dice
Kristine 2*2*2=8
Differences Between Laspeyres and Paasche Indices
No. 7x -4y is simplified from 4x + (3y + 3x) -7y
is it 3×y ?
J, combine like terms 7x-4y
im not good at math so would this help me
yes
Asali
I'm not good at math so would you help me
Samantha
what is the problem that i will help you to self with?
Asali
what's the easiest and fastest way to the synthesize AgNP?
China
Cied
types of nano material
I start with an easy one. carbon nanotubes woven into a long filament like a string
Porter
many many of nanotubes
Porter
what is the k.e before it land
Yasmin
what is the function of carbon nanotubes?
Cesar
what is nanomaterials​ and their applications of sensors.
what is nano technology
what is system testing?
preparation of nanomaterial
Yes, Nanotechnology has a very fast field of applications and their is always something new to do with it...
what is system testing
what is the application of nanotechnology?
Stotaw
In this morden time nanotechnology used in many field . 1-Electronics-manufacturad IC ,RAM,MRAM,solar panel etc 2-Helth and Medical-Nanomedicine,Drug Dilivery for cancer treatment etc 3- Atomobile -MEMS, Coating on car etc. and may other field for details you can check at Google
Azam
anybody can imagine what will be happen after 100 years from now in nano tech world
Prasenjit
after 100 year this will be not nanotechnology maybe this technology name will be change . maybe aftet 100 year . we work on electron lable practically about its properties and behaviour by the different instruments
Azam
name doesn't matter , whatever it will be change... I'm taking about effect on circumstances of the microscopic world
Prasenjit
how hard could it be to apply nanotechnology against viral infections such HIV or Ebola?
Damian
silver nanoparticles could handle the job?
Damian
not now but maybe in future only AgNP maybe any other nanomaterials
Azam
can nanotechnology change the direction of the face of the world
At high concentrations (>0.01 M), the relation between absorptivity coefficient and absorbance is no longer linear. This is due to the electrostatic interactions between the quantum dots in close proximity. If the concentration of the solution is high, another effect that is seen is the scattering of light from the large number of quantum dots. This assumption only works at low concentrations of the analyte. Presence of stray light.
the Beer law works very well for dilute solutions but fails for very high concentrations. why?
how did you get the value of 2000N.What calculations are needed to arrive at it
Privacy Information Security Software Version 1.1a
Good
how do you translate this in Algebraic Expressions
Need to simplify the expresin. 3/7 (x+y)-1/7 (x-1)=
. After 3 months on a diet, Lisa had lost 12% of her original weight. She lost 21 pounds. What was Lisa's original weight?