copyright © 2015, 2011, 2007 pearson education, inc. 1 1 chapter 8 rational exponents, radicals,...
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Copyright © 2015, 2011, 2007 Pearson Education, Inc. 1Copyright © 2015, 2011, 2007 Pearson Education, Inc. 1
Chapter 8
Rational Exponents,
Radicals, and Complex Numbers
Copyright © 2015, 2011, 2007 Pearson Education, Inc. 2
Rational Exponents, Radicals, and Complex Numbers
8.1 Radical Expressions and Functions8.2 Rational Exponents8.3 Multiplying, Dividing, and Simplifying
Radicals8.4 Adding, Subtracting, and Multiplying
Radical Expressions8.5 Rationalizing Numerators and
Denominators of Radical Expressions8.6 Radical Equations and Problem Solving 8.7 Complex Numbers
CHAPTER
8
Copyright © 2015, 2011, 2007 Pearson Education, Inc. 3
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.
8.2
Copyright © 2015, 2011, 2007 Pearson Education, Inc. 4
Rational exponent: An exponent that is a rational number.
Rational Exponents with a Numerator of 1
a1/n = where n is a natural number other than 1.,n a
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.
Copyright © 2015, 2011, 2007 Pearson Education, Inc. 5
Example
Rewrite using radicals, then simplify if possible. a. 491/2 b. 6251/4 c. (216)1/3
Solution
a.
b.
c.
1/ 249
1/4625
1/3216
49 7
4 625 5
3 216 6
Copyright © 2015, 2011, 2007 Pearson Education, Inc. 6
continued
Rewrite using radicals, then simplify. d. (16)1/4 e. 491/2 f. y1/6
Solution
d.
e.
f.
1/4( 16)
1/249
1/6y
4 16 There is no real number answer.
49 7
6 y
Copyright © 2015, 2011, 2007 Pearson Education, Inc. 7
continued
Rewrite using radicals, then simplify. g. (100x8)1/2 h. 9y1/5 i.
Solution
d.
e.
f.
8 1/2(100 )x
1/59y1/28
49
w
8 4100 10x x
59 y
8 4
49 7
w w
1/28
49
w
Copyright © 2015, 2011, 2007 Pearson Education, Inc. 8
General Rule for Rational Exponents
where a 0 and m and n are natural numbers other than 1.
/ ,m
nm n m na a a
Copyright © 2015, 2011, 2007 Pearson Education, Inc. 9
Example
Rewrite using radicals, then simplify, if possible. a. 272/3 b. 2433/5 c. 95/2
Solutiona.
b.
c.
2/3 1/3 227 (27 )
3/5 1/5 3243 (243 )
5/2 1/2 59 (9 )
23( 27) 23 9
35( 243) 33 27
5(3) 243 5( 9)
Copyright © 2015, 2011, 2007 Pearson Education, Inc. 10
continued
Rewrite using radicals, then simplify, if possible. d. e. f.
Solutiond.
e.
f.
33/21 1
16 16
52/5 2x x
3/5 35(4 1) (4 1)x x
31
4
1
64
3/21
16
2/5x 3/5(4 1)x
Copyright © 2015, 2011, 2007 Pearson Education, Inc. 11
Negative Rational Exponents
where a 0, and m and n are natural numbers with n 1.
//
1,m n
m na
a
Copyright © 2015, 2011, 2007 Pearson Education, Inc. 12
Example
Rewrite using radicals; then simplify if possible. a. 251/2 b. 272/3
Solutiona.
b.
1/ 21/ 2
125
25
23
1
27
1 1
525
2/32/3
127
27 2
1 1
3 9
Copyright © 2015, 2011, 2007 Pearson Education, Inc. 13
continued
Rewrite using radicals; then simplify if possible. c. d.
Solutionc.
1/2
1/2
25 1
36 25
36
2/3
1
( 27)
1
2536
2/3d. ( 27)
23
1
( 27)
1/225
36
156
6
5
2/3( 27)
2
1
( 3)
1
9
Copyright © 2015, 2011, 2007 Pearson Education, Inc. 14
Example
Write each of the following in exponential form.
a.
Solution
6 5x
6 5x 5/ 6x
b. 34
1
x
a.
b. 34
1
x
3/4
1
x3/4x
Copyright © 2015, 2011, 2007 Pearson Education, Inc. 15
continued
Write each of the following in exponential form.
c.
Solution
45 x
45 x 4/5x
d. 34 5 2x
c.
d. 34 5 2x 3/ 45 2x
Copyright © 2015, 2011, 2007 Pearson Education, Inc. 16
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.Negative exponents:
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:
1 , n
n
aa
m n m na a a
m n m na a a
nm mna a
n n nab a b
n na bb a
n
n
na ab b
1 ,n
n
aa
Copyright © 2015, 2011, 2007 Pearson Education, Inc. 17
Example
Use the rules of exponents to simplify. Write the answer with positive exponents.
Solution
3/ 4 1/ 4y y
3/ 4 1/ 4y y 3/ 4 ( 1/ 4)y 2/ 4y1/ 2y
Use the product rule for exponents. (Add the exponents.)
Add the exponents.
Simplify the rational exponent.
Copyright © 2015, 2011, 2007 Pearson Education, Inc. 18
Example
Use the rules of exponents to simplify. Write the answer with positive exponents.
Solution
1/3 1/63 4a a
1/3 1/63 4a a 1/3 1/612a 2/6 1/612a
3/6 1/212 or 12a a
Use the product rule for exponents. (Add the exponents.)
Rewrite the exponents with a common denominator of 6.
Add the exponents.
Copyright © 2015, 2011, 2007 Pearson Education, Inc. 19
Example
Use the rules of exponents to simplify. Write the answer with positive exponents.
Solution
Use the quotient for exponents. (Subtract the exponents.)
Rewrite the subtraction as addition.
Add the exponents.
5/ 6
1/ 6
y
y
5/ 6
1/ 6
y
y 5/ 6 ( 1/ 6)y
5/ 6 1/ 6y
y
Copyright © 2015, 2011, 2007 Pearson Education, Inc. 20
Example
Use the rules of exponents to simplify. Write the answer with positive exponents.
Solution
Add the exponents.
2/5 3/53 5y y
2/5 3/53 5y y 2/5 3/515y
1/515y
Copyright © 2015, 2011, 2007 Pearson Education, Inc. 21
Example
Use the rules of exponents to simplify. Write the answer with positive exponents.
Solution
27/8m
27/8m (7/8)2m7/4m
Copyright © 2015, 2011, 2007 Pearson Education, Inc. 22
Example
Use the rules of exponents to simplify. Write the answer with positive exponents.
Solution
32/5 4/53a b
32/5 4/53a b 3 2/5 3 4/5 33 ( ) ( )a b
(2/5)3 (4/5)327a b
6/5 12/527a b
Copyright © 2015, 2011, 2007 Pearson Education, Inc. 23
Example
Use the rules of exponents to simplify. Write the answer with positive exponents.
Solution
8/3 3
6
(2 )x
x
8/3 3
6
(2 )x
x
3 8/3 3
6
2 ( )x
x
8
6
8x
x
8 68x 28x
Copyright © 2015, 2011, 2007 Pearson Education, Inc. 24
Example
Rewrite as a radical with a smaller root index. Assume that all variables represent nonnegative values.
a. b.
Solution
4 64
4a. 64 1/4642 1/4(8 )
21/48 1/28
6 10x
8
6 10b. x 10/6x5/3x
3 5x
3 2x x
Copyright © 2015, 2011, 2007 Pearson Education, Inc. 25
continued
Rewrite as a radical with a smaller root index. Assume that all variables represent nonnegative values.
c.
Solution
6 28 w y
6 28c. w y6 2 1/8( )w y
61/8 21/8w y
3/4 1/4w y
1/43w y
34 w y
Copyright © 2015, 2011, 2007 Pearson Education, Inc. 26
ExamplePerform the indicated operations. Write the result
using a radical.
Solution
b.a. 34x x
a. 34x x 1/ 2 3/ 4x x 1/ 2 3/ 4x 2/ 4 3/ 4x 5/ 4x
54 x
6 7
3
x
x
b.6 7
3
x
x
7 / 6
1/3
x
x
7 / 6 1/3x 7 / 6 2/ 6x
5/ 6x6 5x
Copyright © 2015, 2011, 2007 Pearson Education, Inc. 27
continuedPerform the indicated operations. Write the result
using a radical.
Solution
c. 45 4
c. 45 4 1/2 1/45 4 2/4 1/45 4
1/425 4 1/4(25 4)
1/41004 100
Copyright © 2015, 2011, 2007 Pearson Education, Inc. 28
Example
Write the expression below as a single radical. Assume that all variables represent nonnegative values.
Solution
4 x
4 x 1/2 1/4( )x(1/2)(1/4)x1/8x
8 x
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