LESSON 6.1 EXPONENTS

CONCEPT 1:PROPERTIES OF EXPONENTS

Exponential NotationExponents are used to indicate repeated multiplication of the samenumber.

For example, we use exponential notation to write:

5 � 5 � 5 � 5 � 54

54 is read “five to the fourth power.”

In the expression 54:

• The base, 5, is the repeated factor.

• The exponent, 4, indicates the number of times the base appears as a factor. An exponent is also called a power.

54 � 5 � 5 � 5 � 5 � 625 4 factors Product

Exponent

Base �

LESSON 6.1 EXPONENTS EXPLAIN 367

Concept 1 has sections on

• Exponential Notation

• Multiplication Property

• Division Property

• Power of a Power Property

• Power of a ProductProperty

• Power of a QuotientProperty

• Zero Power Property

• Using Several Properties ofExponents

LESSON 6.1EXPONENTS

Overview

Rosa plans to invest $1000 in an Individual Retirement Account (IRA).She can invest in bonds that offer a return of 7% annually, or a riskierstock fund that is expected to return 10% annually.

Rosa would like to know how much her money can grow in 30 years.Exponents can help her answer this question.

In this lesson, you will study exponents and their properties.

Explain

Find: 23

Solution

The base is 2 and the exponent is 3.

Rewrite using exponential notation: 10 � 10 � 10 � 10 � 10 � 10

Solution

There are six factors. Each is 10.Therefore, the base is 10 and the exponent is 6.

Exponents have several properties. We will use these properties to simplifyexpressions.

In the properties that follow, each variable represents a real number.

Multiplication Property

a. Use the Multiplication Property of Exponents to simplify 23 � 24.

b. Use the definition of exponential notation to justify your answer.

Solution

a. The operation is multiplication and the bases are the same. Therefore, add the exponents and use 2 as the base.

23 � 24 � 23 � 4 � 27

b. Rewrite the product to show the factors. Then simplify.

23 � 24 = (2 � 2 � 2) � (2 � 2 � 2 � 2) = 2 � 2 � 2 � 2 � 2 � 2 � 2 = 27 �

3 factors

� �

4 factors 7 factors

Example 6.1.3

— Property —Multiplication Property of Exponents

English To multiply two exponential expressions with the samebase, add their exponents. The base stays the same.

Algebra xm � xn � xm � n

(Here, m and n are positive integers.)

Example 54 � 52 � 54 � 2 � 56

10 � 10 � 10 � 10 � 10 � 10 = 106 �

6 factors

Example 6.1.2

23 = 2 � 2 � 2 = 8 �

3 factors

Example 6.1.1

368 TOPIC 6 EXPONENTS AND POLYNOMIALS

Remember to add the exponents, but leave the bases alone. That is, 23 � 24 � 23 � 4 � 27, not 47.

Note the difference between 23 � 24 and 23 � 24.

23 � 24 � 23 � 4 � 27 � 12823 � 24 � 8 � 16 � 24

If possible, use the Multiplication Property of Exponents to simplify eachexpression:

a. (�2)2 � (�2)4 b. �22 � 24 c. �22 � (�2)4

Solution

a. In (�2)2 � (�2)4, the base (�2)2 � (�2)4 � (�2)2 � 4

is �2. � (�2)6

� 64

b. In �22 � 24, the base is 2. �22 � 24 � �(22) � (24)We may think of �22 � 24 � �(22 � 4)as the opposite of 22 � 24. � �(26)

� �64

c. In �22 � (�2)4, the base of the first factor, �22, is 2.

The base of the second factor, (�2)4, is �2.

The bases are not the same, so we cannot use the MultiplicationProperty of Exponents.

However, we can still evaluate the expression. �22 � (�2)4 � �4 � 16 � �64

We can extend the Multiplication Property of Exponents to multiply morethan two factors.

Find: 84 � 8 � 85. Leave your answer in exponential notation.

Solution

The bases are the same, so we can use the Multiplication Property of Exponents.

Note: 8 � 81 84 � 8 � 85 � 84 � 81 � 85 � 84 � 1 � 5 � 810

Example 6.1.5

Example 6.1.4

— Caution —Negative Bases

A negative sign is part of the base only when the negative sign isinside the parentheses that enclose the base.

For example, consider the following cases:

In (�3)2, the base is �3. In �32, the base is 3 .

(�3)2 � (�3) � (�3) � �9 You can think of �32 as the “opposite” of 32.

�32 � �(3 � 3 ) � �9


We left 810 in exponential form. Toevaluate 810, use the “yx” key on ascientific calculator or the “^” key on a graphing calculator.

810 � 1,073,741,824

Find: x7 � x3 � x5

Solution

The operation is multiplication andthe bases are the same. Therefore, add the exponents and x7 � x3 � x5 � x7 � 3 � 5 � x15

use x as the base.

Division Property

a. Use the Division Property of Exponents to find �55

3

4�.


Solution

a. The bases are the same, �55

3

4�¬� �54

1� 3� � �

15

�

so subtract the exponents.

b. Rewrite the numerator and �55

3

4�¬� �5

5� 5

� 5� 5

� 5� 5

�

denominator to show the factors.

Cancel the common factors. ¬� �51�

51�� 5

1�� 5

1�� 5

1�� 5

1�� 5

� � �15

�

Example 6.1.7

— Property —Division Property of Exponents

English To divide two exponential expressions with the same base:Compare the exponents.

• If the greater exponent is in the numerator, write the base in the numerator.

• If the greater base is in the denominator, write the base in the denominator.

Then subtract the smaller exponent from the greater. Use the result as the new exponent.

Algebra Example

�xx

m

n� � xm � n for m � n and x � 0. �22

5

3� � 25 � 3 � 22

�xx

m

n� � �xn

1� m� for m � n and x � 0. �

22

3

5� � �25

1� 3� � �

212�


Example 6.1.6


Since 3 � 4, we use the form �xx

m

n� � �xn

1� m� .

Find: 79 76. Leave your answer in exponential notation.

Solution

The operation is division and the bases are the same. Therefore, subtract 79 76 � �

77

9

6� � 79 � 6 � 73

the exponents and use 7 as the base.

Find: w8 w13

Solution

The operation is division and the bases are the same. Therefore, subtract w8 w13 � �

ww

1

8

3� � �w13

1� 8� � �

w1

5�

the exponents and use w as the base.

Power of a Power Property

a. Use the Power of a Power Property of Exponents to simplify (52)3.


Solution

a. To raise a power to a power, (52)3 � 52 � 3 � 56

multiply the exponents.

b. Rewrite each power to show the factors. Then simplify.

( 52)3 = ( 52) � ( 52) � ( 52) = (5 � 5) � (5 � 5) � (5 � 5) = 56�

3 factors

�

6 factors

Example 6.1.10

— Property —Power of a Power Property of Exponents

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

Algebra (xm)n � xmn


Example (72)4 � 72 � 4 � 78

Example 6.1.9

Example 6.1.8



m

n� � xm � n.


m

n� � �xn

1� m� .

Simplify: (y5)3

Solution

To simplify a power of a power, (y5)3 � y5 � 3 � y15

multiply the exponents.

Power of a Product Property

a. Use the Power of a Product Property of Exponents to simplify (3y)2.


Solution

a. Raise each factor to the power 2. (3y)2¬� 32y2 � 9y2

b. Rewrite the power to show (3y)2¬� (3y) � (3y) the factors. Then simplify. ¬� 3 � 3 � y � y

¬� 32y2

¬� 9y2

Simplify: (23 � w5)4

Solution

Use the Power of a Product Property of Exponents to raise each factor inside (23 � w5)4¬� (23)4(w5)4

the parentheses to the power 4.

Use the Power of a Power Property ¬� (23 � 4)(w5 � 4)of Exponents.

Simplify. ¬� 212w20

Example 6.1.13

Example 6.1.12

— Property —Power of a Product Property of Exponents

English To raise a product to a power, you can first raise eachfactor to the power. Then multiply.

Algebra (xy)n � xnyn (Here, n is a positive integer.)

Example (2x)3 � 23x3 � 8x3

Example 6.1.11


We left 212 in exponential form. Toevaluate 212, use the “yx” key on ascientific calculator or the “^” key on a graphing calculator.

212 � 4096

Power of a Quotient Property

a. Use the Power of a Quotient Property of Exponents to simplify ��25

��3

.


Solution

a. Raise the numerator to the power 3. ��25

��3

� �25

3

3� � �1825�

Raise the denominator to the power 3.

b. Rewrite the power to show the factors. Then simplify.

��25

��3

� ��25

�� 25

�� 25

�� 25

�

�

25

�

�

25

� � �25

3

3� � �1825�

Zero Power Property

Here’s a way to understand why 170 is 1.

Suppose we write 0 as 2 � 2.Then, 170 � 172 � 2.

By the Division Property of Exponents, 172 � 2 � �1177

2

2� � �1171�71�

�

�

117

1�

71�

� � 1.

Since 170 � 172 � 2 and 172 � 2 � 1, we have 170 � 1.

— Property —Zero Power Property

English Any real number, except zero, raised to the power 0 is 1.

Algebra x0 � 1, x � 0

Example 170 � 1

Example 6.1.14

— Property —Power of a Quotient Property of Exponents

English To raise a quotient to a power, you can first raise thenumerator and denominator each to the power. Thendivide.

Algebra ��xy

��n

� �xy

n

n�, y � 0

(Here, n is a positive integer.)

Example ��2x

��4

� �2x4

4� � �

1x64�


This same reasoning applies no matterwhat power or nonzero base we choose.

�xx

n

n� � xn � n � x0

�xx

n

n� � 1

Therefore, x0 � 1 for x � 0.

�

3 factors

a. Use the Zero Power Property to simplify 50.

b. Justify your answer.

Solution

a. Any real number, except zero, 50¬� 1raised to the power 0 is 1.

b. Suppose we have �55

3

3�.

We can simplify this using the �55

3

3�¬� 53 � 3 � 50

Division Property of Exponents.

But if we reduce the fraction �55

3

3�, �55

3

3�¬� �51�51�

�

�

51�51� �

�

51�51�

� � 1 the result is 1.

Since �55

3

3� is equivalent to both 50 and 1,

we conclude 50 � 1.

Find each of the following. (Assume each variable represents a nonzeroreal number).

a. (�7)0 b. �w4

0� c. (12x4y5)0 d. �2y0 e. 00

Solution

In each case, we apply the Zero Power Property: any nonzero real numberraised to the zero power is 1.

a. The base is the real number �7. (�7)0¬� 1

b. The base, w, represents a nonzero real number. �w4

0�¬� �

14

�

c. The base, 12x4y5, represents a nonzero (12x4y5)0¬� 1real number.

d. Only y is raised to the power 0. �2y0 � �2 � 1¬� �2

e. In the Zero Power Property, 00 is undefinedthe base cannot be 0.

Using Several Properties of ExponentsTo simplify an exponential expression, we may need to use severalproperties of exponents.

Example 6.1.16

Example 6.1.15


Find: ��25

2�8x4

��3

Solution

First, we simplify the expression � ��25

2�8x4

��3

inside the parentheses.

To combine the powers of 2, subtract exponents. � ��

2x4

3��3

(Division Property of Exponents)

Raise the numerator and the denominator each to the power 3. � �

((2x4

3))

3

3�

(Power of a Quotient Property of Exponents)

Multiply exponents: 4 � 3 � 12 and 3 � 3 � 9. � �2x4

3

�

�

3

3�

(Power of a Power Property of Exponents)

� �x2

1

9

2�

Find: �(x3

(�

yx4)

5

2y4)3

�

Solution

First, we simplify the expression � �(x3

(�

yx4)

5

2y4)3

�

inside the parentheses in the numerator.

To combine the powers of x ,add their exponents. � �

(x(y

8y4)

4

2)3

�

(Multiplication Property of Exponents)

In the numerator, raise each factor to the power 3. � �

(x8

()y

3

4()y2

4)3�

(Power of a Product Property of Exponents)

Multiply exponents: 8 � 3 � 24 and 4 � 3 = 12 and 4 � 2 � 8. � �

x2

y

4y8

12�

(Power of a Power Property of Exponents).

To combine the powers of y, subtract their exponents. � x24y4

(Division Property of Exponents)

Example 6.1.18

Example 6.1.17



m

n� � �xn

1� m�.

Since 12 > 8, we use the form �xx

m

n� � xm � n.

Find: �2x6

2(x3yy5

4)2�

Solution

Raise each factor inside the � �2x

6

2(x3yy5

4)2�

parentheses to the power 2.(Power of a Product Property of Exponents) � �

2x2(63x)y

2(5y4)2

�

To simplify (y4)2, multiply exponents: 4 � 2 � 8. � �

2x6

2

x3y

2

5y8

�

(Power of a Power Property of Exponents)

Multiply the constants: 2 � 32 � 2 � 9 � 18 � �168xxy

2

5y8

�

Divide 18 by 6. To combine the powers of x, subtract their exponents. � 3xy3

To combine the powers of y, subtract their exponents.(Division Property of Exponents)

Real world problems often involve exponents. For example, the followingformula may be used to calculate the value of an investment after a certainnumber of years.

A � P(1 � r)t

where A is the value of the investment, P is the original principal invested, r is the annual rate of return, and t is the number of years the money is invested.

Rosa plans to invest $1000 in an Individual Retirement Account (IRA).She can invest in a bond fund that averages a 7% annual return, or in ariskier stock fund that is expected to have a 10% annual return.

a. Determine the value of the bond fund after 30 years.

b. Determine the projected value of the stock fund after 30 years.

c. Compare the returns on the two investments.

Example 6.1.20

Example 6.1.19


Solution

For each investment, the principal, A � P(1 � r)t

P, is $1000. The time, t, is 30 years.

a. For the bond fund, the annual rate A � 1000(1 � 0.07)30

of return is 7%. So, r � 0.07.In the formula, substitute 1000 for P, 0.07 for r, and 30 for t.

Add 1 and 0.07. � 1000(1.07)30

On a calculator, use the “yx ” key � 1000(7.612255043)or the “^” key to approximate 1.0730.

Multiply and round to the nearest � $7,612.26hundredth (cent).

After 30 years, the bond fund will be worth $7,612.26.

b. For the stock fund, the projected annual A � 1000(1 � 0.10)30

rate of return is 10%. So r � 0.10. In the formula, substitute 1000 for P, 0.10 for r, and 30 for t.

Add 1 and 0.10. � 1000(1.10)30

On a calculator, use the “yx ” key or � 1000(17.44940227)the “^” key to approximate 1.1030.

Multiply and round to the nearest � $17,449.40hundredth (cent).

After 30 years, the stock fund should be worth $17,449.40.

c. The bond fund would grow to almost 8 times its original value.

The stock fund would grow to over 17 times its original value.

The stock fund, which is riskier than the bond fund, is projected to be worth more than twice as much as the bond fund in 30 years.


To get a better estimate, we waited untilthe end of the problem to round theanswer.

Here is a summary of this concept from Interactive Mathematics.


LESSON 6.1 EXPONENTS CHECKLIST 379

exponential notationbase

exponentpower

Ideas and Procedures❶ Exponential Notation

Given an expression written in exponential Example 6.1.1notation, identify the base, identify the Find: 23

exponent, and evaluate the expression. See also: Example 6.1.2

❷ Properties of ExponentsUse the following properties of exponents to Example 6.1.18simplify an expression:

Multiplication Property of Exponents Find: �(x3

(�

yx4)

5

2y4)3

�

Division Property of ExponentsPower of a Power Property of Exponents See also: Example 6.1.3-6.1.17, 6.1.19, 6.1.20Power of a Product Property of Exponents Apply 1-28Power of a Quotient Property of ExponentsZero Power Property

Checklist Lesson 6.1Here is what you should know after completing this lesson.

Words and Phrases

Homework

Homework Problems

Circle the homework problems assigned to you by the computer, then complete them below.

ExplainProperties of ExponentsUse the appropriate properties of exponents to simplifythe expressions in questions 1 through 12. (Keep youranswers in exponential form where possible.)

1. Find:

a. 32 � 35 b. 52 � 55

c. 72 � 75

2. Find:

a. �33

9

5� b. �33

5

9�

c. �33

9

9�

3. Find:

a. (73)2 b. (72)3

4. Find:

a. (5 � x)3 b. (3 � y)2

c. (a2 � b)4

5. Find:

a. ��x3

x�4x5

��2

b. ��aa1

9

2

�

�

aa7

6��

4

c. ��bb6

3�

�

bb

5

8��3

d. �22

3

5�

�

xx

5

2�

6. Find:

a. (a2 � a3)2 � (a2 � a3)2

b. �y4 �

y83y2�

c. x4 � x9 � x � y5 � y11

7. Find:

a. (b3)2 � (b4)3

b. �yy1

6

7� � (y5)2 � (y3)4

c. �aa1

4

1�

�

bb

6

3�

8. Find:

a. �y(9x�

y)x

4

7� b. �((33bb2))

6

4�

9. As animals grow, they get taller faster than they getstronger. In general, this proportion of increase in

height to increase in strength can be written as �xx

2

3�.

Simplify this fraction.

10. An animal is proportionally stronger the smaller itis. If a person is 200 times as tall as an ant, figureout how much stronger a person is, pound for

pound, by simplifying the expression �220000

2

3�.

11. Find:

a. ��54xx2yy

2

zz3��

0

b. �yy9

7

�

�

yy2�

c. ��bb3

6�

�

bb

5

3��4

d. �2x0 � 5y0

12. Find:

a. ��(x3

x�

7x4)2��

5

b. �(4a2)0

2� 3b0�

c. ��(33x11�

�

3xx

2

7)2

��3

d. ��(b2b�

8

b7)��

4


LESSON 6.1 EXPONENTS APPLY 381

Apply

Practice Problems

Here are some additional practice problems for you to try.

Properties of Exponents1. Find: 75 � 73. Leave your answer in exponential

notation.

2. Find: 63 � 64. Leave your answer in exponentialnotation.

3. Find: b12 � b3

4. Find: c9 � c4

5. Find: a6 � a5

6. Find: 57 53. Leave your answer in exponentialnotation.

7. Find: 910 94. Leave your answer in exponentialnotation.

8. Find: �mm

1

4

0�

9. Find: �nn

2

1

0

5�

10. Find: �bb

1

5

2�

11. Find: (53)4. Leave your answer in exponentialnotation.



14. Find: (y8)3

15. Find: (z12)4

16. Find: (x9)4

17. Find: (3 � a)4

18. Find: (4 � b)2

19. Find: (2 � y)3

20. Find: �aa

6

8bb

5

2�

21. Find: �mm

3

7

nn1

4

0�

22. Find: �xx

3yy

7

8zz

1

5

2�

23. Find: 50

24. Find: 3480

25. Find: x0

26. Find: 51 � (4z)0

27. Find: a0 � (xyz)0 � 31

28. Find: 21 � (3x)0 � y0

Evaluate

Practice Test

Take this practice test to be sure that you are prepared for the final quiz in Evaluate.


1. Rewrite each expression below. Keep your answerin exponential form where possible.

a. 11 � 11 � 11 � 11

b. 3 � 3 � y � y � y � y � y

c. 512 � 58 � 523

d. x7 � y � y19 � x14 � y6

e. 78 � b5 � b8 � 710 � b

2. Rewrite each expression below in simplest formusing exponents.

a. �2 � 22� 2

� 2� 2

� 2� 2 � 2

�

b. �bb

2

1

0

4�

c. �33

1

9

2

�

�

xx1

7

6�

d. �y14 �

yy

17

3 � y4�

3. Circle the expressions below that simplify to �xy

3

5�.

�xx

6

3yy

2

7� �yy

1

2

1

xx4

5�

�xx6yy

9

4� �xx4

7

yy6�

4. Circle the expressions below that simplify to 5y.

(31x8)0 � 5y

�(�5y)0

�5yy2�

�(5

5yy)2

�

5. Simplify each expression below.

a. (b4 � b2)8

b. (35 � a6)2

c. (29 � x4 � y6)11

6. Simplify each expression below.

a. ��53yx

1

8

0��

4

b. ��75aa

3b2

4��

6

7. Calculate the value of each expression below.

a. (4x)0 � 2y0

b. (5xy2 � 4x3)0

c. �2x0 � y0

d. �(42x)0� � �

32x0� � �

�

22x0�

8. Rewrite each expression below using a singleexponent.

a. ��aa4

�

�

aa3

5��

7

b. ��aa4�

�

aa

3

5��7

5 � 5 � 5 � y � y � y � y��

5 � 5 � y � y

Lesson 6.1 ExponentsHomework1a. 37 b. 57 c. 77 3a. 76 b. 76

5a. x 8 b. a8 c. 1 d. 7a. b 18 b. y 11 c.

9. 11a. 1 b. c. d. 3

Apply - Practice Problems1. 78 3. b 15 5. a 11 7. 96 9. n 5

11. 512 13. 1330 15. z 48 17. 81a 4

19. 8y 3 21. 23. 1 25. 1 27. 3

Evaluate - Practice Test1a. 114 b. 32y 5 c. 543 d. x 21y 26 e. 718 b14

2a. 23 b. b6 c. d.

3. and 4. (31x 8)0 � 5y, , and

5a. b48 b. 310a12 c. 299 x 44y 66

6a. b.

7a. –1 b. 1 c. –3 d. 1

8a. a 35 b.

Lesson 6.2 Polynomial Operations IHomework

1. 3 y 3 + 3y 2 – 5 3a. –4y 5 – 2y 3 + 3y + 2

b. 5, 3, 1, 0 c. 5 5. –4v7 + v 3 + 6v2 – 5v + 5

7. –7s3t 3 + 7st 2 – s2t + 2st –13t + 9

9. 2x 2y + 10xy 2 + 4y 3 + 3

11. 4w 2yz + 3w 3 – 4wyz 2 + 6wyz – 4wy 2z + 3

13. x 3y 3z 3 15. –3t 4u 4v 15 17. 10p 3r 4 + 5p4r 5

19. 21. 23. + or (4 + 5xy )xy 2�

35x 2y 3�

34xy 2�

33a 3d�2b 5c 3

3xw 5�

y

1�4

1�a35

76a6b24�

5654y 40�34x 32

(5y )2�

5y5y 2�

yx7y�x 4y 6

x 6y 2�x 3y 7

1�y 4

33�x9

m 4�n 6

1�b4

1�y 3

1�x

b3�a7

x3�

4

LESSON 6.2: ANSWERS 727

Apply - Practice Problems1. 2xy � 5xz ; 9y 2 � 13yz – 8z 2

3a.binomial b. binomial c. trinomial d. monomial

3e. trinomial

5. 8 7. 9 9. 7 11. 6 13. 84 15. 6x 2 � 11x – 8

17. 15m 2n 3 � 2m 2n 2 – 7mn 19. 15a 3b 2 � 4a 2b – ab 3

21. 20xy 2z 3 – 30x 2yz 2 � 10x 3y 3z 23. 4x 3 � 7x – 8

25. y 2 � 6xy � 4y 27. 11a 5b 3 – 4a 4b – 9b 29. 15y 5

31. –45a 9 33. 28x 4y 8 35. –6w 2x 5y 3z 3

37. –6a 7b 7 � 10a 4b 5 – 12a 4b 2

39. 20a 4b 2 � 10a 4b 3 – 35a 3b 4 – 15a 2b 3

41. 12x 6y 3 – 28x 4y 5 � 8x 4y 4 – 4x 3y 4

43. 5a 2b 5 45. 47. 49.

51. 4a � 3a 3 53. � 4x 3y 55. –

Evaluate - Practice Test 1. t 2 – s + 5, m5n4o3p2r, and c15 + c11 – 3�

2. w 5x 4 is a monomial.

2x 2 – 36 is a binomial.

x17 + x12 – is a trinomial.

27 is a monomial.

27x 3 – 2x 2y 3 is a binomial.

x 2 + 3xy – y 2 is a trinomial.

3. 8w 8 + 7w 5 + 3w 3 – 13w 2 – 2

4a. 3x 3y – 8x 2y 2 – 5y 3 + xy + y 2 + 19

b. 7x 3y – 8x 2y 2 + 3y 3 + 5xy – y 2 + 7

5. x 8y 3w 5

6. 3n3p3 + 2n5p5 – 35n2p 7

7.

8. – t 41�2

3t 2u�

2v

3x 4yz 6�

2

2�3

1�3

2�3

1�3

3�14

5�7

x 2z 3�2y 2

2�x

7�y

3n 5p 3�

2mq3x 3y 2z 5�

2w8a 2b 3�

3c

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LESSON 6.1 EXPONENTS