CONCEPT 1: PROPERTIES OF EXPONENTS Exponential Notation Exponents are used to indicate repeated multiplication of the same number. For example, we use exponential notation to write: 5 5 5 5 5 4 5 4 is read “five to the fourth power.” In the expression 5 4 : • 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. 5 4 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 Product Property • Power of a Quotient Property • Zero Power Property • Using Several Properties of Exponents LESSON 6.1 EXPONENTS 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 riskier stock 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
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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.
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
376 TOPIC 6 EXPONENTS AND POLYNOMIALS
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.
LESSON 6.1 EXPONENTS EXPLAIN 377
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.
378 TOPIC 6 EXPONENTS AND POLYNOMIALS
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
380 TOPIC 6 EXPONENTS AND POLYNOMIALS
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.
12. Find: (82)5. Leave your answer in exponentialnotation.
13. Find: (135)6. 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.
382 TOPIC 6 EXPONENTS AND POLYNOMIALS
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