Download Chapter 8 Rational Exponents, Radicals, and

Chapter 8
Rational
Exponents,
Radicals, and
Complex
Numbers
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
11
CHAPTER
8
Rational Exponents, Radicals,
and Complex Numbers
8.1
8.2
8.3
8.4
8.5
8.6
8.7
Radical Expressions and Functions
Rational Exponents
Multiplying, Dividing, and Simplifying
Radicals
Adding, Subtracting, and Multiplying
Radical Expressions
Rationalizing Numerators and
Denominators of Radical Expressions
Radical Equations and Problem Solving
Complex Numbers
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8.2
Rational Exponents
1. Evaluate rational exponents.
2. Write radicals as expressions raised to
rational exponents.
3. Simplify expressions with rational number
exponents using the rules of exponents.
4. Use rational exponents to simplify radical
expressions.
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Rational exponent: An exponent that is a rational
number.
Rational Exponents with a Numerator of 1
a1/n =
n
a ,where n is a natural number other than 1.
Note: If a is negative and n is odd, then the root is negative.
If a is negative and n is even, then there is no real number root.
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Example
Rewrite using radicals, then simplify if possible.
a. 491/2
b. 6251/4
c. (216)1/3
Solution
1/ 2
a. 49  49  7
b. 6251/4  4 625  5
c. 2161/3  3 216  6
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continued
Rewrite using radicals, then simplify.
d. (16)1/4
e. 491/2
f. y1/6
Solution
1/4
(

16)
 4 16  There is no real number answer.
d.
e. 491/2   49  7
f. y1/6  6 y
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continued
Rewrite using radicals, then simplify. 1/2
8
8
1/2
1/5

w
g. (100x )
h. 9y
i.  
 49 
Solution
8 1/2
(100
x
)  100 x8  10 x4
d.
e. 9y1/5  9
1/2
w 
f.  
 49 
8

5
y
w8 w4

49
7
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General Rule for Rational Exponents
a
m/ n
 a 
n
m
 a
n
m
, where a  0 and m and n are
natural numbers other than 1.
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Example
Rewrite using radicals, then simplify, if possible.
a. 272/3
b. 2433/5
c. 95/2
Solution
2/ 3
1/ 3 2
a. 27  (27 )  ( 3 27 ) 2  32  9
b. 2433/ 5  (2431/ 5 )3  ( 5 243)3  33  27
c. 95/2  (91/2 )5  ( 9)5  (3)5  243
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continued
Rewrite3/2using radicals, then simplify, if possible.
d.  1 
e. x 2/5
f. (4 x  1)3/5
 16 
Solution
3
3
3/2
d.  1    1    1   1


 
64
4

16
 16 


e. x2/5  5 x2
f. (4 x  1)3/5  5 (4 x  1)3
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Negative Rational Exponents
a
m / n

1
a
m/n
, where a  0, and m and n are natural
numbers with n  1.
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Example
Rewrite using radicals; then simplify if possible.
a. 251/2
b. 272/3
Solution
a. 251/ 2 
b. 27
2 / 3
1
1
1


1/ 2
25
25 5
1
 2/3 
27

1
3
27

2
1 1
 2 
3
9
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continued
Rewrite using radicals; then simplify if possible.
1/2
c.  25 
d. (27) 2/3
 
 36 
Solution
1/2
c.  25 
 
 36 


d. (27) 2/3 
1
1/2
 25 
 
 36 
1
1
6


5
25
5
6
36
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1
2/3
(27)
1
 3
( 27 ) 2
1
1

2 
(3)
9
13
Example
Write each of the following in exponential form.
a.
6
x
5
b.
1
4
x3
Solution
6
a.
b.
5/ 6

x
x
5
1
 3/4  x 3/4
x
x3
1
4
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continued
Write each of the following in exponential form.
c.
 x
5
4
d.
4
 5x  2
3
Solution
c.
d.
 x  x
5
4
4
4/5
 5x  2   5 x  2 
3
3/ 4
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Rules of Exponents Summary
(Assume that no denominators are 0, that a and b are
real numbers, and that m and n are integers.)
Zero as an exponent:
a0 = 1, where a  0.
00 is indeterminate.
n
n
n
n
1
1
a
b
Negative exponents:

a
,
a  , a
b a
an
Product rule for exponents:
Quotient rule for exponents:
Raising a power to a power:
Raising a product to a power:
Raising a quotient to a power:
n
a m a n  a mn
a m  a n  a mn
m n
mn
a

a
 
n n
ab

a
b
 
a n
an
 b   bn
n
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Example
Use the rules of exponents to simplify. Write the
answer with positive exponents. y 3/ 4  y 1/ 4
Solution
y
3/ 4
y
1/ 4
y
3/ 4 ( 1/ 4)
Use the product rule for exponents.
(Add the exponents.)
 y 2/ 4
Add the exponents.
 y1/ 2
Simplify the rational exponent.
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Example
Use the rules of exponents to simplify. Write the
answer with positive exponents.  3a1/3  4a1/6 
Solution
 3a  4a 
1/3
1/6
 12a
Use the product rule for exponents.
(Add the exponents.)
 12a 2/61/6
Rewrite the exponents with a
common denominator of 6.
1/31/6
 12a 3/6 or 12a1/2
Add the exponents.
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Example
Use the rules of exponents to simplify. Write the
answer with positive exponents. y5/ 6
y 1/ 6
Solution
y 5/ 6
y 1/ 6
y
5/ 6( 1/ 6)
Use the quotient for exponents.
(Subtract the exponents.)
 y 5/ 61/ 6
Rewrite the subtraction as addition.
y
Add the exponents.
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Example
Use the rules of exponents to simplify. Write the
answer with positive exponents.
2/5
3/5

3
y
5
y

 
Solution
 3 y  5 y   15y
2/5
3/5
2/53/5
 15y1/5
Add the exponents.
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Example
Use the rules of exponents to simplify. Write the
answer with positive exponents.
m 
7/8 2
Solution
m 
7/8 2
 m (7/8)2
 m 7/4
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Example
Use the rules of exponents to simplify. Write the
answer with positive exponents.
3a

2/5 4/5 3
b
Solution
3a b

2/5 4/5 3
 33 (a 2/5 )3 (b4/5 )3
 27a (2/5)3b(4/5)3
 27a 6/5b12/5
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Example
Use the rules of exponents to simplify. Write the
answer with positive exponents.
8/3 3
(2 x )
x6
Solution
(2 x8/3 )3 23 ( x8/3 )3

6
x
x6
8x8
 6
x
 8x86
 8x 2
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Example
Rewrite as a radical with a smaller root index. Assume
that all variables represent nonnegative values.
a. 4 64
b. 6 x10
Solution
a. 4 64  641/4
b.
6
x10
 x10/6
 (8 )
 x 5/3
 821/4
 3 x5
2 1/4
 81/2
 8
 x 3 x2
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continued
Rewrite as a radical with a smaller root index. Assume
that all variables represent nonnegative values.
c. 8 w6 y 2
Solution
c.
8
6
w y
2
 ( w6 y 2 )1/8


 w61/8
y 21/8
 w3/4 y1/4
 w y
3
1/4
 4 w3 y
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Example
Perform the indicated operations. Write the result
using a radical.
6 7
x
3
4
a. x  x
b. 3
Solution
a.
x  4 x3  x1/ 2  x 3/ 4
 x1/ 2  3/ 4
 x 2 / 4  3/ 4
 x5 / 4
 4 x5
x
6
7
7/6
x
x
b.
 1/ 3
3
x
x
 x 7 / 6 1/ 3
 x 7 / 62 / 6
 x5 / 6
 6 x5
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continued
Perform the indicated operations. Write the result
using a radical.
c.
54 4
Solution
c.
5  4 4  51/2  41/4
 52/4  41/4
 5  4
2
1/4
 (25  4)1/4
 1001/4
 4 100
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Example
Write the expression below as a single radical. Assume
that all variables represent nonnegative values.
4
Solution
4
x
x  ( x1/2 )1/4
 x (1/2)(1/4)
 x1/8
8x
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