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
Concept 1 has sections on
• Exponential Notation
CONCEPT 1:
PROPERTIES OF EXPONENTS
• Multiplication Property
• Division Property
Exponential Notation
• Power of a Power Property
Exponents are used to indicate repeated multiplication of the same
number.
• Power of a Product
Property
For example, we use exponential notation to write:
• Power of a Quotient
Property
54 is read “five to the fourth power.”
• Zero Power Property
In the expression 54:
• Using Several Properties of
Exponents
•
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 5 5 5 54
Exponent
54 5 5 5 5 625
Base
4 factors
LESSON 6.1 EXPONENTS
Product
EXPLAIN
367
Example 6.1.1
Find: 23
Solution
23 = 2 2 2 = 8
3 factors
The base is 2 and the exponent is 3.
Example 6.1.2
Rewrite using exponential notation: 10 10 10 10 10 10
Solution
10 10 10 10 10 10 = 106
There are six factors. Each is 10.
Therefore, the base is 10 and the
exponent is 6.
6 factors
Exponents have several properties. We will use these properties to simplify
expressions.
In the properties that follow, each variable represents a real number.
Multiplication Property
— Property —
Multiplication Property of Exponents
English To multiply two exponential expressions with the same
base, 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 5 6
Example 6.1.3
a. Use the Multiplication Property of Exponents to simplify 23 24.
b. Use the definition of exponential notation to justify your answer.
Solution
Remember to add the exponents,
but leave the bases alone.
That is, 23 24 23 4 27, not 47.
a. The operation is multiplication and the bases are the same.
Therefore, add the exponents and use 2 as the base.
Note the difference between
23 24 and 23 24.
b. Rewrite the product to show the factors. Then simplify.
368
23 24 = (2 2 2) (2 2 2 2) = 2 2 2 2 2 2 2 = 27
TOPIC 6 EXPONENTS AND POLYNOMIALS
3 factors
27
24
23 4
128
23 2 4 8 16 24
23
23 24 23 4 27
4 factors
7 factors
— Caution —
Negative Bases
A negative sign is part of the base only when the negative sign is
inside 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
Example 6.1.4
If possible, use the Multiplication Property of Exponents to simplify each
expression:
a. (2)2 (2)4
b. 22 24
c. 22 (2)4
Solution
a. In (2)2 (2)4, the base
is 2.
b. In 22 24, the base is 2.
We may think of 22 24
as the opposite of 22 24.
(2)2 (2)4 (2)2 4
(2)6
64
22 24 (22) (24)
(22 4)
(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 Multiplication
Property 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 more
than two factors.
Example 6.1.5
Find: 84 8 85. Leave your answer in exponential notation.
We left 810 in exponential form. To
evaluate 810, use the “yx ” key on a
scientific calculator or the “^” key
on a graphing calculator.
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
LESSON 6.1 EXPONENTS
810 1,073,741,824
EXPLAIN
369
Example 6.1.6
Find: x7 x3 x5
Solution
The operation is multiplication and
the bases are the same.
Therefore, add the exponents and
use x as the base.
x7 x3 x5 x7 3 5 x15
Division Property
— 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
xm
xm n for m n and x 0.
xn
xm
1
for m n and x 0.
xn
xn m
25
3 25 3 22
2
23
1
1
5 2
2
25 3
2
(Here, m and n are positive integers.)
Example 6.1.7
xm
x
1
.
Since 3 4, we use the form n nm
x
53
5
a. Use the Division Property of Exponents to find 4 .
b. Use the definition of exponential notation to justify your answer.
Solution
a. The bases are the same,
so subtract the exponents.
b. Rewrite the numerator and
denominator to show the factors.
53
1
1
4 ¬ 5
5
54 3
53
555
4 ¬ 5
5555
1
Cancel the common factors.
1
370
TOPIC 6 EXPONENTS AND POLYNOMIALS
1
1
5 5 5
5 5 5 5
1
5
¬ 1
1
Example 6.1.8
Find: 79 76. Leave your answer in exponential notation.
Solution
The operation is division and the
bases are the same. Therefore, subtract
the exponents and use 7 as the base.
xm
x
79
7
Since 9 6, we use the form n x m n.
79 76 6 79 6 73
Example 6.1.9
Find: w8 w13
Solution
The operation is division and the
bases are the same. Therefore, subtract
the exponents and use w as the base.
w8
w
1
1
w
5
w8 w13 13 13 8
w
xm
x
1
.
Since 8 13, we use the form n nm
x
Power of a Power Property
— Property —
Power of a Power Property of Exponents
English To raise a power to a power, multiply the exponents.
Algebra (xm)n xmn
(Here, m and n are positive integers.)
Example (72)4 72 4 78
Example 6.1.10
a. Use the Power of a Power Property of Exponents to simplify (52)3.
b. Use the definition of exponential notation to justify your answer.
Solution
a. To raise a power to a power,
multiply the exponents.
(52)3 52 3 56
b. Rewrite each power to show the factors. Then simplify.
= ( 52 ) ( 52 ) ( 52 ) = (5 5) (5 5) (5 5) = 56
3
( 52 )
3 factors
6 factors
LESSON 6.1 EXPONENTS
EXPLAIN
371
Example 6.1.11
Simplify: (y5)3
Solution
To simplify a power of a power,
multiply the exponents.
(y5)3 y5 3 y15
Power of a Product Property
— Property —
Power of a Product Property of Exponents
English To raise a product to a power, you can first raise each
factor to the power. Then multiply.
Algebra (xy)n xnyn
Example
(2x)3
23x3
(Here, n is a positive integer.)
8x3
Example 6.1.12
a. Use the Power of a Product Property of Exponents to simplify (3y)2.
b. Use the definition of exponential notation to justify your answer.
Solution
a. Raise each factor to the power 2.
(3y)2¬ 32y2 9y2
b. Rewrite the power to show
the factors. Then simplify.
(3y)2¬ (3y) (3y)
¬ 3 3 y y
¬ 32y2
¬ 9y2
Example 6.1.13
Simplify: (23 w5)4
Solution
Use the Power of a Product Property
of Exponents to raise each factor inside
the parentheses to the power 4.
We left 212 in exponential form. To
evaluate 212, use the “yx ” key on a
scientific calculator or the “^” key
on a graphing calculator.
Use the Power of a Power Property
of Exponents.
¬ (23 4)(w5 4)
Simplify.
¬ 212w20
2 12 4096
372
(23 w5)4¬ (23)4(w5)4
TOPIC 6 EXPONENTS AND POLYNOMIALS
Power of a Quotient Property
— Property —
Power of a Quotient Property of Exponents
English To raise a quotient to a power, you can first raise the
numerator and denominator each to the power. Then
divide.
n
xy xn
y
Algebra n , y 0
(Here, n is a positive integer.)
4
2x 24
x
16
x
Example 4 4
Example 6.1.14
3
25 a. Use the Power of a Quotient Property of Exponents to simplify .
b. Use the definition of exponential notation to justify your answer.
Solution
3
a. Raise the numerator to the power 3.
Raise the denominator to the power 3.
25
23
5
8
125
3 b. Rewrite the power to show the factors.
Then simplify.
3
222
2
8
25 25 25 555
5
125
3
3
2
5
3 factors
Zero Power Property
— 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
Here’s a way to understand why 170 is 1.
This same reasoning applies no matter
what power or nonzero base we choose.
Suppose we write 0 as 2 2.
Then, 170 172 2.
172
17
1
xn
n xn n x0
x
xn
n 1
x
1
1
7
17
17 17
By the Division Property of Exponents, 172 2 2 1.
Since 170 172 2 and 172 2 1, we have 170 1.
1
1
Therefore, x0 1 for x 0.
LESSON 6.1 EXPONENTS
EXPLAIN
373
Example 6.1.15
a. Use the Zero Power Property to simplify 50.
b. Justify your answer.
Solution
50¬ 1
a. Any real number, except zero,
raised to the power 0 is 1.
53
5
b. Suppose we have 3 .
53
3 ¬ 53 3 50
5
We can simplify this using the
Division Property of Exponents.
1
1
1
1
1
1
53
5 5 5
3 ¬ 1
5
5 5 5
53
But if we reduce the fraction 3 ,
5
the result is 1.
53
Since 3 is equivalent to both 5 0 and 1,
5
we conclude 50 1.
Example 6.1.16
Find each of the following. (Assume each variable represents a nonzero
real number).
a. (7)0
w0
4
b. c. (12x4y5) 0
d. 2y0
e. 00
Solution
In each case, we apply the Zero Power Property: any nonzero real number
raised 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.
c. The base, 12x4y5, represents a nonzero
real number.
d. Only y is raised to the power 0.
w0
1
¬ 4
4
(12x4y5)0¬ 1
2y0 2 1¬ 2
e. In the Zero Power Property,
the base cannot be 0.
00 is undefined
Using Several Properties of Exponents
To simplify an exponential expression, we may need to use several
properties of exponents.
374
TOPIC 6 EXPONENTS AND POLYNOMIALS
Example 6.1.17
4 3
2 2 x 5
Find: 8
Solution
First, we simplify the expression
inside the parentheses.
4 3
2 2 x 5
8
To combine the powers of 2,
subtract exponents.
(Division Property of Exponents)
3
Raise the numerator and the
denominator each to the power 3.
(Power of a Quotient Property of Exponents)
33
Multiply exponents: 4 3 12 and 3 3 9.
(Power of a Power Property of Exponents)
3 3
4 3
2x xm
x
1
.
Since 5 8, we use the form n nm
x
(x4)3
(2 )
x4 3
2
x12
2
9
Example 6.1.18
(x3 x5y4)3
(y )
Find: 4
2
Solution
First, we simplify the expression
inside the parentheses in the numerator.
(x3 x5y4)3
(y )
4
2
To combine the powers of x ,
add their exponents.
(Multiplication Property of Exponents)
4
2
In the numerator, raise each
factor to the power 3.
(Power of a Product Property of Exponents)
4
2
Multiply exponents: 8 3 24
and 4 3 = 12 and 4 2 8.
(Power of a Power Property of Exponents).
8
To combine the powers of y,
subtract their exponents.
(Division Property of Exponents)
x 24y 4
(x8y4)3
(y )
(x8)3(y4)3
(y )
x24y12
y
xm
x
Since 12 > 8, we use the form n x m n.
LESSON 6.1 EXPONENTS
EXPLAIN
375
Example 6.1.19
2x2(3y4)2
6xy
Find: 5
Solution
Raise each factor inside the
parentheses to the power 2.
(Power of a Product Property of Exponents)
2x2(3y4)2
6xy
2x2(3)2(y4)2
6xy5
5
To simplify (y 4 )2, multiply
exponents: 4 2 8.
(Power of a Power Property of Exponents)
5
Multiply the constants: 2 3 2 2 9 18
5
Divide 18 by 6.
To combine the powers of x, subtract their exponents.
To combine the powers of y, subtract their exponents.
(Division Property of Exponents)
2x232y8
6xy
18x2y8
6xy
3xy3
Real world problems often involve exponents. For example, the following
formula may be used to calculate the value of an investment after a certain
number 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.
Example 6.1.20
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 a
riskier 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.
376
TOPIC 6 EXPONENTS AND POLYNOMIALS
Solution
For each investment, the principal,
P, is $1000. The time, t, is 30 years.
A P(1 r)t
a. For the bond fund, the annual rate
of return is 7%. So, r 0.07.
In the formula, substitute 1000 for P,
0.07 for r, and 30 for t.
A 1000(1 0.07)30
Add 1 and 0.07.
1000(1.07)30
On a calculator, use the “yx ” key
or the “^” key to approximate 1.0730.
1000(7.612255043)
Multiply and round to the nearest
hundredth (cent).
$7,612.26
To get a better estimate, we waited until
the end of the problem to round the
answer.
After 30 years, the bond fund will be worth $7,612.26.
b. For the stock fund, the projected annual
rate of return is 10%. So r 0.10.
In the formula, substitute 1000 for P,
0.10 for r, and 30 for t.
A 1000(1 0.10)30
Add 1 and 0.10.
1000(1.10)30
On a calculator, use the “yx ” key or
the “^” key to approximate 1.1030.
1000(17.44940227)
Multiply and round to the nearest
hundredth (cent).
$17,449.40
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
Here is a summary of this concept from Interactive Mathematics.
378
TOPIC 6 EXPONENTS AND POLYNOMIALS
Checklist Lesson 6.1
Here is what you should know after completing this lesson.
Words and Phrases
exponential notation
base
exponent
power
Ideas and Procedures
❶ Exponential Notation
Given an expression written in exponential
notation, identify the base, identify the
exponent, and evaluate the expression.
❷ Properties of Exponents
Use the following properties of exponents to
simplify an expression:
Multiplication Property of Exponents
Division Property of Exponents
Power of a Power Property of Exponents
Power of a Product Property of Exponents
Power of a Quotient Property of Exponents
Zero Power Property
Example 6.1.1
Find: 23
See also: Example 6.1.2
Example 6.1.18
(x3 x5y4)3
(y )
Find: 4
2
See also: Example 6.1.3-6.1.17, 6.1.19, 6.1.20
Apply 1-28
LESSON 6.1 EXPONENTS
CHECKLIST
379
Homework
Homework Problems
Circle the homework problems assigned to you by the computer, then complete them below.
7. Find:
Explain
a. (b 3 )2 (b 4 )3
Properties of Exponents
Use the appropriate properties of exponents to simplify
the expressions in questions 1 through 12. (Keep your
answers in exponential form where possible.)
y6
y
a4 b6
c. a11 b3
b. 17 (y 5)2 (y 3)4
8. Find:
1. Find:
a.
32
35
c.
72
75
b.
52
(xy)4
y x
x2
x
height to increase in strength can be written as 3 .
35
b. 9
3
Simplify this fraction.
39
c. 9
10. An animal is proportionally stronger the smaller it
is. If a person is 200 times as tall as an ant, figure
out how much stronger a person is, pound for
3
3. Find:
a. (7 3)2
2002
200
b. (7 2)3
pound, by simplifying the expression 3 .
4. Find:
a. (5 11. Find:
x) 3
b. (3 y) 2
2
5. Find:
3
5 2
6
5
3
8
x x x 3
b b
c. b b
a. 4
aa
a6
a
3
2 x5
d. 25 x2
12
b. 7
9
4
0
54xxyyzz 4
b b
c. b b
a. 23
c. (a 2 b) 4
3
5
6
3
a. (a 2 a 3 )2 (a 2 a 3 )2
y4 3y2
y
b. 8
c. x4 x9 x y5 y11
TOPIC 6 EXPONENTS AND POLYNOMIALS
y7 y
y y
2
b. 9
d. 2x 0 5y 0
12. Find:
4 2 5
(x x x ) 3
a. 7
2 2 3
6. Find:
380
b. 2 4
9. As animals grow, they get taller faster than they get
stronger. In general, this proportion of increase in
2. Find:
39
a. 5
3
(3b)6
(3b )
7
a. 9
55
(33x 3xx ) c. 11
7
(4a2)0 3b0
2
b. 4
(b b b ) 8
d. 2
7
Apply
Practice Problems
Here are some additional practice problems for you to try.
Properties of Exponents
14. Find: (y 8)3
1. Find: 7 5 7 3. Leave your answer in exponential
notation.
15. Find: (z 12)4
2. Find: 6 3 6 4. Leave your answer in exponential
notation.
16. Find: (x 9)4
17. Find: (3 a) 4
3. Find: b 12 b 3
18. Find: (4 b) 2
4. Find: c 9 c 4
19. Find: (2 y) 3
5. Find: a 6 a 5
20. Find: 82
6. Find: 5 7 5 3. Leave your answer in exponential
notation.
7. Find: 9 10 9 4. Leave your answer in exponential
notation.
m10
m
n20
9. Find: 15
n
b12
10. Find: b5
8. Find: 4
11. Find: (53)4. Leave your answer in exponential
notation.
a6b5
ab
m7n4
21. Find: 3 m n10
x3y7z12
22. Find: 8
xy z5
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
12. Find: (82)5. Leave your answer in exponential
notation.
13. Find: (135)6. Leave your answer in exponential
notation.
LESSON 6.1 EXPONENTS
APPLY
381
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 answer
in exponential form where possible.
a. 11 11 11 11
c.
58
c. (2 9 x 4 y 6)11
5 23
6. Simplify each expression below.
d. x 7 y y 19 x 14 y 6
e.
78
b5
b8
7 10
10 4
b
2
a. (4x) 0 2y 0
b. (5xy 2 4x 3) 0
c. 2x 0 y 0
(4x)0
2
3x0
2
2x0
2
d. x3
3. Circle the expressions below that simplify to 5 .
y
y11x5
2
y x4
x7y
46
xy
4. Circle the expressions below that simplify to 5y.
(31x 8) 0 5y
(5y) 0
5y2
y
(5y)2
5y
555yyyy
55yy
382
3 4
7. Calculate the value of each expression below.
222222
222
2
0
b
b. 14
b
312 x7
c. 39 x16
y17
d. 14
y y3 y4
a. 37
xy
xy9
64
xy
53yx 6
7a b
b. 5a
a. 8
2. Rewrite each expression below in simplest form
using exponents.
x6y2
a. (b 4 b 2)8
b. (3 5 a 6)2
b. 3 3 y y y y y
5 12
5. Simplify each expression below.
TOPIC 6 EXPONENTS AND POLYNOMIALS
8. Rewrite each expression below using a single
exponent.
5 7
aa aa 7
aa
b. a a
4
a. 3
3
4
5