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This module is from Elementary Algebra by Denny Burzynski and Wade Ellis, Jr. The symbols, notations, and properties of numbers that form the basis of algebra, as well as exponents and the rules of exponents, are introduced in this chapter. Each property of real numbers and the rules of exponents are expressed both symbolically and literally. Literal explanations are included because symbolic explanations alone may be difficult for a student to interpret.Objectives of this module: understand the power rules for powers, products, and quotients.

Overview

  • The Power Rule for Powers
  • The Power Rule for Products
  • The Power Rule for quotients

The power rule for powers

The following examples suggest a rule for raising a power to a power:

( a 2 ) 3 = a 2 a 2 a 2

Using the product rule we get

( a 2 ) 3 = a 2 + 2 + 2 ( a 2 ) 3 = a 3 2 ( a 2 ) 3 = a 6

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( x 9 ) 4 = x 9 x 9 x 9 x 9 ( x 9 ) 4 = x 9 + 9 + 9 + 9 ( x 9 ) 4 = x 4 9 ( x 9 ) 4 = x 36

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Power rule for powers

If x is a real number and n and m are natural numbers,
( x n ) m = x n m

To raise a power to a power, multiply the exponents.

Sample set a

Simplify each expression using the power rule for powers. All exponents are natural numbers.

( x 3 ) 4 = x 3 4 x 12 The box represents a step done mentally.

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( y 5 ) 3 = y 5 3 = y 15

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( d 20 ) 6 = d 20 6 = d 120

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( x ) = x

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Although we don’t know exactly what number is, the notation indicates the multiplication.

Practice set a

Simplify each expression using the power rule for powers.

The power rule for products

The following examples suggest a rule for raising a product to a power:

( a b ) 3 = a b a b a b Use the commutative property of multiplication . = a a a b b b = a 3 b 3

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( x y ) 5 = x y x y x y x y x y = x x x x x y y y y y = x 5 y 5

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( 4 x y z ) 2 = 4 x y z 4 x y z = 4 4 x x y y z z = 16 x 2 y 2 z 2

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Power rule for products

If x and y are real numbers and n is a natural number,
( x y ) n = x n y n

To raise a product to a power, apply the exponent to each and every factor.

Sample set b

Make use of either or both the power rule for products and power rule for powers to simplify each expression.

( a x y ) 4 = a 4 x 4 y 4

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( 3 a b ) 2 = 3 2 a 2 b 2 = 9 a 2 b 2 Don't forget to apply the exponent to the 3!

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( 2 s t ) 5 = 2 5 s 5 t 5 = 32 s 5 t 5

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( a b 3 ) 2 = a 2 ( b 3 ) 2 = a 2 b 6 We used two rules here . First, the power rule for products . Second, the power rule for powers.

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( 7 a 4 b 2 c 8 ) 2 = 7 2 ( a 4 ) 2 ( b 2 ) 2 ( c 8 ) 2 = 49 a 8 b 4 c 16

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If 6 a 3 c 7 0 , then ( 6 a 3 c 7 ) 0 = 1 Recall that x 0 = 1 for x 0.

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[ 2 ( x + 1 ) 4 ] 6 = 2 6 ( x + 1 ) 24 = 64 ( x + 1 ) 24

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Practice set b

Make use of either or both the power rule for products and the power rule for powers to simplify each expression.

( 3 b x y ) 2

9 b 2 x 2 y 2

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[ 4 t ( s 5 ) ] 3

64 t 3 ( s 5 ) 3

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( 9 x 3 y 5 ) 2

81 x 6 y 10

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( 1 a 5 b 8 c 3 d ) 6

a 30 b 48 c 18 d 6

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[ ( a + 8 ) ( a + 5 ) ] 4

( a + 8 ) 4 ( a + 5 ) 4

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[ ( 12 c 4 u 3 ( w 3 ) 2 ] 5

12 5 c 20 u 15 ( w 3 ) 10

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[ 10 t 4 y 7 j 3 d 2 v 6 n 4 g 8 ( 2 k ) 17 ] 4

10 4 t 16 y 28 j 12 d 8 v 24 n 16 g 32 ( 2 k ) 68

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( x 3 x 5 y 2 y 6 ) 9

( x 8 y 8 ) 9 = x 72 y 72

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( 10 6 10 12 10 5 ) 10

10 230

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The power rule for quotients

The following example suggests a rule for raising a quotient to a power.

( a b ) 3 = a b a b a b = a a a b b b = a 3 b 3

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Power rule for quotients

If x and y are real numbers and n is a natural number,
( x y ) n = x n y n , y 0

To raise a quotient to a power, distribute the exponent to both the numerator and denominator.

Sample set c

Make use of the power rule for quotients, the power rule for products, the power rule for powers, or a combination of these rules to simplify each expression. All exponents are natural numbers.

( 2 x b ) 4 = ( 2 x ) 4 b 4 = 2 4 x 4 b 4 = 16 x 4 b 4

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( a 3 b 5 ) 7 = ( a 3 ) 7 ( b 5 ) 7 = a 21 b 35

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( 3 c 4 r 2 2 3 g 5 ) 3 = 3 3 c 12 r 6 2 9 g 15 = 27 c 12 r 6 2 9 g 15 or 27 c 12 r 6 512 g 15

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[ ( a 2 ) ( a + 7 ) ] 4 = ( a 2 ) 4 ( a + 7 ) 4

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[ 6 x ( 4 x ) 4 2 a ( y 4 ) 6 ] 2 = 6 2 x 2 ( 4 x ) 8 2 2 a 2 ( y 4 ) 12 = 36 x 2 ( 4 x ) 8 4 a 2 ( y 4 ) 12 = 9 x 2 ( 4 x ) 8 a 2 ( y 4 ) 12

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( a 3 b 5 a 2 b ) 3 = ( a 3 2 b 5 1 ) 3 We can simplify within the parentheses . We have a rule that tells us to proceed this way . = ( a b 4 ) 3 = a 3 b 12 ( a 3 b 5 a 2 b ) 3 = a 9 b 15 a 6 b 3 = a 9 6 b 15 3 = a 3 b 12 We could have actually used the power rule for quotients first . Distribute the exponent, then simplify using the other rules . It is probably better, for the sake of consistency, to work inside the parentheses first.

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( a r b s c t ) w = a r w b s w c t w

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Practice set c

Make use of the power rule for quotients, the power rule for products, the power rule for powers, or a combination of these rules to simplify each expression.

( 2 x 3 y ) 3

8 x 3 27 y 3

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( x 2 y 4 z 7 a 5 b ) 9

x 18 y 36 z 63 a 45 b 9

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[ 2 a 4 ( b 1 ) 3 b 3 ( c + 6 ) ] 4

16 a 16 ( b 1 ) 4 81 b 12 ( c + 6 ) 4

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( 8 a 3 b 2 c 6 4 a 2 b ) 3

8 a 3 b 3 c 18

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[ ( 9 + w ) 2 ( 3 + w ) 5 ] 10

( 9 + w ) 20 ( 3 + w ) 50

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[ 5 x 4 ( y + 1 ) 5 x 4 ( y + 1 ) ] 6

1 , if x 4 ( y + 1 ) 0

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( 16 x 3 v 4 c 7 12 x 2 v c 6 ) 0

1 , if x 2 v c 6 0

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Exercises

Use the power rules for exponents to simplify the following problems. Assume that all bases are nonzero and that all variable exponents are natural numbers.

( 10 a 2 b ) 2

100 a 4 b 2

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( x 2 y 3 z 5 ) 4

x 8 y 12 z 20

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( x 3 y 2 z 4 ) 5

x 15 y 10 z 20

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( a 4 b 7 c 6 d 8 ) 8

a 32 b 56 c 48 d 64

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( 1 8 c 10 d 8 e 4 f 9 ) 2

1 64 c 20 d 16 e 8 f 18

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( x y ) 4 ( x 2 y 4 )

x 6 y 8

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( a 2 b 3 ) 3 ( a 3 b 3 ) 4

a 18 b 21

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( h 3 k 5 ) 2 ( h 2 k 4 ) 3

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( x 4 y 3 z ) 4 ( x 5 y z 2 ) 2

x 26 y 14 z 8

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( a b 3 c 2 ) 5 ( a 2 b 2 c ) 2

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( 6 a 2 b 8 ) 2 ( 3 a b 5 ) 2

4 a 2 b 6

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( a 3 b 4 ) 5 ( a 4 b 4 ) 3

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( x 6 y 5 ) 3 ( x 2 y 3 ) 5

x 8

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( a 8 b 10 ) 3 ( a 7 b 5 ) 3

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( m 5 n 6 p 4 ) 4 ( m 4 n 5 p ) 4

m 4 n 4 p 12

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( x 8 y 3 z 2 ) 5 ( x 6 y z ) 6

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( 10 x 4 y 5 z 11 ) 3 ( x y 2 ) 4

1000 x 8 y 7 z 33

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( 9 a 4 b 5 ) ( 2 b 2 c ) ( 3 a 3 b ) ( 6 b c )

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( 2 x 3 y 3 ) 4 ( 5 x 6 y 8 ) 2 ( 4 x 5 y 3 ) 2

25 x 14 y 22

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( 3 a b 4 x y ) 3

27 a 3 b 3 64 x 3 y 3

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( 3 a 2 b 3 c 4 ) 3

27 a 6 b 9 c 12

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( 4 2 a 3 b 7 b 5 c 4 ) 2

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[ x 2 ( y 1 ) 3 ( x + 6 ) ] 4

x 8 ( y 1 ) 12 ( x + 6 ) 4

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( x n + 2 ) 3 x 2 n

x n + 6

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Exercises for review

( [link] ) Is there a smallest integer? If so, what is it?

no

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( [link] ) Use the distributive property to expand 5 a ( 2 x + 8 ) .

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( [link] ) Find the value of ( 5 3 ) 2 + ( 5 + 4 ) 3 + 2 4 2 2 5 1 .

147

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( [link] ) Assuming the bases are not zero, find the value of ( 4 a 2 b 3 ) ( 5 a b 4 ) .

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( [link] ) Assuming the bases are not zero, find the value of 36 x 10 y 8 z 3 w 0 9 x 5 y 2 z .

4 x 5 y 6 z 2

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Questions & Answers

what is the meaning of function in economics
Effah Reply
Pls, I need more explanation on price Elasticity of Supply
Isaac Reply
Is the degree to the degree of responsiveness of a change in quantity supplied of goods to a change in price
Afran
Discuss the short-term and long-term balance positions of the firm in the monopoly market?
Rabindranath Reply
hey
Soumya
hi
Mitiku
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Mitiku
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Soumya
through hard study and performing well than expected from you
Mitiku
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Mitiku
parna kya hai behencho?
Soumya
Hallo
Rabindranath
Hello, dear what's up?
Mitiku
cool
Momoh
good morning
Isaac
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Isaac
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Afran
Ys please I'm in Ghana
Afran
what is firms
Anteyi Reply
A firm is a business entity which engages in the production of goods and aimed at making profit.
Avuwada
What is autarky in Economics.
Avuwada
what is choice
Tia Reply
So how is the perfect competition different from others
Rev Reply
what is choice
Tia
please what type of commodity is 1.Beaf 2.Suagr 3.Bread
Alfred Reply
1
Naziru
what is the difference between short run and long run?
Ukpen Reply
It just depends on how far you would like to run!!!🤣🤣🤣
Anna
meaning? You guys need not to be playing here; if you don't know a question, leave it for he that knows.
Ukpen
pls is question from which subject or which course
Ada
Is this not economics?
Ukpen
This place is meant to be for serious educational matters n not playing ground so pls let's make it a serious place.
Docky
Is there an economics expert here?
Docky
Okay and I was being serous
Anna
The short run is a period of time in which the quantity of at least one inputs is fixed...
Anna
that is the answer that I found online and in my text book
Anna
Elacisity
salihu
Meaning of economics
Suraj Reply
It will creates rooms for an effective demands.
Chinedum Reply
different between production and supply
babsnof
Hii
Suraj
hlo
eshita
What is the economic?
Suraj
Economics is a science which study human behavior as a relationship between ends and scarce means which has an alternative use.
Mr
what is supply
babsnof
what is different between demand and supply
Debless Reply
Demand refers to the quantity of products that consumers are willing to purchase at various prices per time while Supply has to do with the quantity of products suppliers are willing to supply at various prices per time. find the difference in between
Saye
Please what are the effects of rationing Effect of black market Effects of hoarding
Atty Reply
monoply is amarket structure charecrized by asingle seller and produce a unique product in the market
Cali Reply
yes
Niraj
I want to know wen does the demand curve shift to the right
Nana
demand curve shifts to the right when there's an increase in price of a substitute or increase in income
kin
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kin
what are the factor that change the curve right
Nana
explain the law of supply in simple .....
freshwater
the Law of supply: states that all factor being equal, when the price of a particular goods increase the supply will also increase, as it decreases the supply will also decrease
kin
@Nana the factor that changes or shift the d demand curve to the right is 1) the increase in price of a substitute good or commodity 2) increase in income
kin
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kin
different between bill of exchange n treasure bill
Nana
yes
Ada
so would you tell me what means an apportunity cost plz?
Cali
what is true cost
Akiti
your question isn't correct naadi
Anthonia
define an apportunity cost?
Cali
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Cali
In a simple term, it is an Alternative foregone.
Sule
opportunity cost is the next best value of a scale of preference
Akiti
Both of you are not correct.
Nelly
opportunity cost: is a forgone alternative
kin
Monopoly is where is one producer produces a given product with no close substitute
James
what is income effect?
Qwecou Reply
if you borrow $5000 to buy a car at 12 percent compounded monthly to be repaid over the next 4 year what is monthly payment
Nitish Reply
Researchers demonstrated that the hippocampus functions in memory processing by creating lesions in the hippocampi of rats, which resulted in ________.
Mapo Reply
The formulation of new memories is sometimes called ________, and the process of bringing up old memories is called ________.
Mapo Reply
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Source:  OpenStax, Elementary algebra. OpenStax CNX. May 08, 2009 Download for free at http://cnx.org/content/col10614/1.3
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