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CHAPTER
10
Rational Exponents, Radicals, and
Complex Numbers
10.1
10.2
10.3
10.4
Radical Expressions and Functions
Rational Exponents
Multiplying, Dividing, and Simplifying Radicals
Adding, Subtracting, and Multiplying Radical
Expressions
10.5 Rationalizing Numerators and Denominators of
Radical Expressions
10.6 Radical Equations and Problem Solving
10.7 Complex Numbers
Copyright © 2011 Pearson Education, Inc.
10.1
1.
2.
3.
4.
5.
6.
Radical Expressions and Functions
Find the nth root of a number.
Approximate roots using a calculator.
Simplify radical expressions.
Evaluate radical functions.
Find the domain of radical functions.
Solve applications involving radical functions.
Copyright © 2011 Pearson Education, Inc.
nth root: The number b is an nth root of a number a
if bn = a.
Evaluating nth roots
When evaluating a radical expression n a , the sign of a
and the index n will determine possible outcomes.
If a is nonnegative, then n a  b, where b  0 and
bn = a.
If a is negative and n is even, then there is no realnumber root.
If a is negative and n is odd, then n a  b , where b is
negative and bn = a.
Copyright © 2011 Pearson Education, Inc.
Slide 10- 3
Example 1
Evaluate each root, if possible.
a. 169
Solution 169  13
b.  0.49
Solution  0.49  0.7
c.
100
Solution
is not a real number because
100
there is no real number whose
square is –100.
Copyright © 2011 Pearson Education, Inc.
Slide 10- 4
continued
Evaluate each root, if possible.
d.  144
Solution  144  12
49
e.
144
Solution
f.
3
7
49
49


12
144
144
27
Solution
3
27  3
Copyright © 2011 Pearson Education, Inc.
Slide 10- 5
continued
Evaluate each root, if possible.
g. 3 27
Solution 3 27  3
h.
4
81
Solution
4
81  3
Copyright © 2011 Pearson Education, Inc.
Slide 10- 6
Some roots, like 3 are called irrational because we
cannot express their exact value using rational
numbers. In fact, writing 3 with the radical sign is
the only way we can express its exact value.
However, we can approximate 3 using rational
numbers.
Approximating to two decimal places: 2  1.41
Approximating to three decimal places: 2  1.414
Note: Remember that the symbol,
“approximately equal to.”
Copyright © 2011 Pearson Education, Inc.

, means
Slide 10- 7
Example 2
Approximate the roots using a calculator or table in the
endpapers. Round to three decimal places.
a. 18
Solution
18  4.243
b.  32
Solution  32  5.657
c.
3
56
Solution
3
56  3.826
Copyright © 2011 Pearson Education, Inc.
Slide 10- 8
Example 3
Find the root. Assume variables represent nonnegative
values.
b.
c.
y
Solution
4
36m
6
36 x10
d.
25 y 4
Solution
Solution
y4  y2
Because (y2)2 = y4.
3

6m
36m
6
Because (6m3)2 = 36m6.
36 x10 6 x5
 2
4
25 y
5y
Copyright © 2011 Pearson Education, Inc.
Slide 10- 9
continued
Find the root. Assume variables represent nonnegative
values.
e.
3
f.
4
y
Solution
9
16
81x
Solution
3
y9
4
 y3
4
81x16  3x
Copyright © 2011 Pearson Education, Inc.
Slide 10- 10
Example 4
Find the root. Assume variables represent any real
number.
Solution
14
a. y
b. 36 y
10
Solution
c. (n  3) Solution
2
7
y14  y
5
36 y10  6 y
(n  3) 2  n  3
Copyright © 2011 Pearson Education, Inc.
Slide 10- 11
continued
Find the root. Assume variables represent any real
number.
12
6
49
y

7
y
49
y
Solution
d.
12
e.
3
9
27n
Solution
3
27n9
 3n3
c. 3 ( w  4)3 Solution 3 ( w  4)3  w  4
Copyright © 2011 Pearson Education, Inc.
Slide 10- 12
Radical function: A function containing a radical
expression whose radicand has a variable.
Example 5a
Given f(x) = 5 x  8, find f(3).
Solution
To find f(3), substitute 3 for x and simplify.
f  3  5  3  8  15  8  7
Copyright © 2011 Pearson Education, Inc.
Slide 10- 13
Example 6
Find the domain of each of the following.
a. f  x   x  8
Solution Since the index is even, the radicand x  8  0
x 8
must be nonnegative.
Domain:  x x  8 , or [8, )
b. f  x   3x  9
Solution The radicand must be nonnegative. 3x  9  0
3x  9
Domain:  x x  3 , or (,3]
x3
Conclusion The domain of a radical function with an even
index must contain values that keep its radicand
nonnegative.
Copyright © 2011 Pearson Education, Inc.
Slide 10- 14
Example 7
If you drop an object, the time (t) it takes in seconds
to fall d feet is given by t  16d . Find the time it
takes for an object to fall 800 feet.
Understand We are to find the time it takes for an
object to fall 800 feet.
Plan Use the formula t  16d , replacing d with 800.
Execute
t
800
16
Replace d with 800.
t  50
Divide within the radical.
t  7.071
Evaluate the square root.
Copyright © 2011 Pearson Education, Inc.
Slide 10- 15
continued
Answer It takes an object 7.071 seconds to fall 800
feet.
Check We can verify the calculations, which we will
leave to the viewer.
Copyright © 2011 Pearson Education, Inc.
Slide 10- 16
For which square root is –12.37 the
approximation for?
a)  3.517
b)
3.517
c)  153
d)
153
Copyright © 2011 Pearson Education, Inc.
Slide 10- 17
For which square root is –12.37 the
approximation for?
a)  3.517
b)
3.517
c)  153
d)
153
Copyright © 2011 Pearson Education, Inc.
Slide 10- 18
Evaluate. 0.0004
a) 0.2
b) 0.02
c) 0.002
d) 0.0002
Copyright © 2011 Pearson Education, Inc.
Slide 10- 19
Evaluate. 0.0004
a) 0.2
b) 0.02
c) 0.002
d) 0.0002
Copyright © 2011 Pearson Education, Inc.
Slide 10- 20
Find the domain of f(x) = 4 x  16 .
a)  x x  4 , or [4, )
b)  x x  4 , or [4, )
c)  x x  4 , or (, 4]
d)  x x  4 , or (, 4]
Copyright © 2011 Pearson Education, Inc.
Slide 10- 21
Find the domain of f(x) = 4 x  16 .
a)  x x  4 , or [4, )
b)  x x  4 , or [4, )
c)  x x  4 , or (, 4]
d)  x x  4 , or (, 4]
Copyright © 2011 Pearson Education, Inc.
Slide 10- 22
10.2
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.
Copyright © 2011 Pearson Education, Inc.
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.
Copyright © 2011 Pearson Education, Inc.
Slide 10- 24
Example 1
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
Copyright © 2011 Pearson Education, Inc.
Slide 10- 25
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
Copyright © 2011 Pearson Education, Inc.
Slide 10- 26
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
Copyright © 2011 Pearson Education, Inc.
Slide 10- 27
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.
Copyright © 2011 Pearson Education, Inc.
Slide 10- 28
Example 2
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
Copyright © 2011 Pearson Education, Inc.
Slide 10- 29
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
Copyright © 2011 Pearson Education, Inc.
Slide 10- 30
Negative Rational Exponents
a
m / n

1
a
m/n
, where a  0, and m and n are natural
numbers with n  1.
Copyright © 2011 Pearson Education, Inc.
Slide 10- 31
Example 3
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
Copyright © 2011 Pearson Education, Inc.
Slide 10- 32
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
Copyright © 2011 Pearson Education, Inc.
1
2/3
(27)
1
 3
( 27 ) 2
1
1

2 
(3)
9
Slide 10- 33
Example 4
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
Copyright © 2011 Pearson Education, Inc.
Slide 10- 34
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
Copyright © 2011 Pearson Education, Inc.
Slide 10- 35
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
n
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:
a m a n  a mn
a m  a n  a mn
m n
mn
a

a
 
Copyright © 2011 Pearson Education, Inc.
n n
ab

a
b
 
a n
an
 b   bn
n
Slide 10- 36
Example 5a
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.
Copyright © 2011 Pearson Education, Inc.
Slide 10- 37
Example 5b
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.
Copyright © 2011 Pearson Education, Inc.
Slide 10- 38
Example 5c
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.
Copyright © 2011 Pearson Education, Inc.
Slide 10- 39
Example 5d
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.
Copyright © 2011 Pearson Education, Inc.
Slide 10- 40
Example 5e
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
Copyright © 2011 Pearson Education, Inc.
Slide 10- 41
Example 5f
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
Copyright © 2011 Pearson Education, Inc.
Slide 10- 42
Example 5g
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
Copyright © 2011 Pearson Education, Inc.
Slide 10- 43
Example 6
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
 x 3 x2
 8
Copyright © 2011 Pearson Education, Inc.
Slide 10- 44
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
Copyright © 2011 Pearson Education, Inc.
Slide 10- 45
Example 7
Perform the indicated operations. Write the result
using a radical.
6
a.
x  4 x3
Solution
a.
x  4 x3  x1/ 2  x 3/ 4
 x1/ 2  3/ 4
 x 2 / 4  3/ 4
x7
b. 3
x
6
7
7/6
x
x
b.
 1/ 3
3
x
x
 x 7 / 6 1/ 3
 x5 / 4
 4 x5
 x 7 / 62 / 6
 x5 / 6
 6 x5
Copyright © 2011 Pearson Education, Inc.
Slide 10- 46
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
Copyright © 2011 Pearson Education, Inc.
Slide 10- 47
Example 8
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
Copyright © 2011 Pearson Education, Inc.
Slide 10- 48
Simplify.  x y
1/ 2

2 / 3 3/ 4
3/ 8 3/ 4
x
y
a)
3/8 1/ 4
x
y
b)
c) x
3/ 4
1/ 2
y
3/8 1/ 2
d) x y
Copyright © 2011 Pearson Education, Inc.
Slide 10- 49
Simplify.  x y
1/ 2

2 / 3 3/ 4
3/ 8 3/ 4
x
y
a)
3/8 1/ 4
x
y
b)
c) x
3/ 4
1/ 2
y
3/8 1/ 2
d) x y
Copyright © 2011 Pearson Education, Inc.
Slide 10- 50
Simplify. 3 25  3 5
a) 5
b) 25
c) 25
d) 5
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Slide 10- 51
Simplify. 3 25  3 5
a) 5
b) 25
c) 25
d) 5
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Slide 10- 52
Simplify.  8 
2 / 3
a) 4
1
b) 4
c) 4
1
d) 
4
Copyright © 2011 Pearson Education, Inc.
Slide 10- 53
Simplify.  8 
2 / 3
a) 4
1
b) 4
c) 4
1
d) 
4
Copyright © 2011 Pearson Education, Inc.
Slide 10- 54
10.3
Multiplying, Dividing, and Simplifying
Radicals
1. Multiply radical expressions.
2. Divide radical expressions.
3. Use the product rule to simplify radical expressions.
Copyright © 2011 Pearson Education, Inc.
Product Rule for Radicals
If both n a and n b are real numbers, then
n
a  n b  n a  b.
Copyright © 2011 Pearson Education, Inc.
Slide 10- 56
Example 1
Find the product and simplify. Assume all variables
represent positive values.
a. 4  9
b. 7  y
Solution
a. 4  9  4  9
b.
7 y
 7y
 36
6
Copyright © 2011 Pearson Education, Inc.
Slide 10- 57
continued
Find the product and simplify. Assume all variables
represent positive values.
c. 4 2  4 8
d. 3 4 x  3 5x
Solution
c.
4
2  4 8  4 2 8
d.
3
4 x  3 5x  3 4 x  5x
 4 16
 3 20x2
2
Copyright © 2011 Pearson Education, Inc.
Slide 10- 58
continued
Find the product and simplify. Assume all variables
represent positive values.
6
y
5
9

e. 7 y  7 y
f.
w 5
Solution
e.
7
y 7 y 
7
y y

7
y14
5
9
5
9
f.
6
y
6 y



w 5
w 5
 y2
Copyright © 2011 Pearson Education, Inc.
6y

5w
Slide 10- 59
continued
Find the product and simplify. Assume all variables
represent positive values.
g. x2  x2
Solution
g.
x2  x2  x2  x2
 x4
 x2
Copyright © 2011 Pearson Education, Inc.
Slide 10- 60
Raising an nth Root to the nth Power
For any nonnegative real number a, n a
 
n
 a.
Quotient Rule for Radicals
If both n a and n b are real numbers, then
n
a na

,
where
b

0.
n
b
b
Copyright © 2011 Pearson Education, Inc.
Slide 10- 61
Example 2
Simplify. Assume variables represent positive values.
a. 11
b. 147
c. 3 15
x6
3
49
Solution
a.
11
11
11


7
49
49
b. 147
3
c.
3
15
x6

3
15
3
x6
3
15
 2
x
147

 49  7
3
Copyright © 2011 Pearson Education, Inc.
Slide 10- 62
continued
Simplify. Assume variables represent positive values.
e. 5 x
d. 5 12
5
1024
4
Solution
5
d. 12  5 12
5
4
4
53
5
5
x
x
x
e. 5
5

1024
1024
4
Copyright © 2011 Pearson Education, Inc.
Slide 10- 63
Simplifying nth Roots
To simplify an nth root,
1. Write the radicand as a product of the greatest
possible perfect nth power and a number or an
expression that has no perfect nth power factors.
2. Use the product rule n ab  n a  n b when a is the
perfect nth power.
3. Find the nth root of the perfect nth power radicand.
Copyright © 2011 Pearson Education, Inc.
Slide 10- 64
Example 3
Simplify.
a. 80
b.
Solution
Solution
6 98  6  49  2
80  16 5
 16
6 98
5
 6  49  2
 67 2
4 5
 42 2
Copyright © 2011 Pearson Education, Inc.
Slide 10- 65
continued
Simplify.
c. 4 3 448
d. 5 4 48
Solution
Solution
4 3 448  4
3
4 4
3
64
3
7
5 4 48  5 4 16  3
7
 5 4 16  4 3
 5 2 4 3
 16 7
3
 10 4 3
Copyright © 2011 Pearson Education, Inc.
Slide 10- 66
Example 4a
Simplify the radical using prime factorization.
Solution
686  7  7  7  2
Write 686 as a product of its prime
factors.
 7 72
The square root of the pair of 7s is 7.
 7 14
Multiply the prime factors in the
radicand.
Copyright © 2011 Pearson Education, Inc.
Slide 10- 67
continued
Simplify the radical using prime factorization.
b. 3 500
c. 4 810
Solution
3
500

2 2555
b.
3
c.
4
810  4 2  3  3  3  3  5
 34 2  5
 53 2  2
 3 4 10
 53 4
Copyright © 2011 Pearson Education, Inc.
Slide 10- 68
Example 5a
Simplify.
32x5
Solution
32x5  16 x4
 16x4
 4 x2 2 x
2x
2x
The greatest perfect square factor of
32x5 is 16x4.
Use the product rule of square roots to
separate the factors into two radicals.
Find the square root of 16x4 and leave
2x in the radical.
Copyright © 2011 Pearson Education, Inc.
Slide 10- 69
Example 5b
Simplify 2 96a4b.
Solution
2 96a b  2 16a
4
 2 16a
 2  4a
4
4
2
 8a 2 6b
6b
6b
The greatest perfect square factor of
96a4b is 16a4.
6b
Use the product rule of square roots to
separate the factors into two radicals.
Find the square root of 16a4 and leave
6b in the radical.
Multiply 2 and 4.
Copyright © 2011 Pearson Education, Inc.
Slide 10- 70
continued
Simplify.
c. y 3 3 y10
d. 5 486x11 y14
Solution
Solution
y
33
y y
10
33
y
33
y y
y 
 y y
3
5
9
9
33
 y6 3 y
3
y
y
486x11 y14
 5 243  2  x10  x  y10  y 4
 5 243  x10  y10  5 2  x  y 4
 3x 2 y 2 5 2 xy 4
Copyright © 2011 Pearson Education, Inc.
Slide 10- 71
Example 6
Find the product or quotient and simplify the results.
Assume that variables represent positive values.
a. 5  8
b. 4 5x5  5 50 x4
Solution
5 8
Solution
 40
4 5x5  5 50 x4
 4 10
 4  5 5x5  50 x4
 2 10
 20 250x9
 20 25 10  x8  x
 20  5 x 4 10 x
 100 x 4 10 x
Copyright © 2011 Pearson Education, Inc.
Slide 10- 72
continued
Find the product or quotient and simplify the results.
Assume that variables represent positive values.
9 6
300
9
245
a
b
c.
d.
4
Solution
300
300

4
4
3 5a 5b
Solution
9 245a 9b6
3 5a 5b
245a9b6
3
5a5b
 3 49a4b5
 75
 3 49a4b4  b
 25  3
 3  7a 2b2 b
5 3
 21a 2b2 b
Copyright © 2011 Pearson Education, Inc.
Slide 10- 73
Simplify. Assume all variables
represent nonnegative numbers.
3m2 9m3
a) 3m 3m
b) 3m
c)
2
3m
27m6
d) 3m2 3m3
Copyright © 2011 Pearson Education, Inc.
Slide 10- 74
Simplify. Assume all variables
represent nonnegative numbers.
3m2 9m3
a) 3m 3m
b) 3m
c)
2
3m
27m6
d) 3m2 3m3
Copyright © 2011 Pearson Education, Inc.
Slide 10- 75
Simplify.
486
a) 6 9
b) 3 54
c) 9 6
d) 18 27
Copyright © 2011 Pearson Education, Inc.
Slide 10- 76
Simplify.
486
a) 6 9
b) 3 54
c) 9 6
d) 18 27
Copyright © 2011 Pearson Education, Inc.
Slide 10- 77
10.4
Adding, Subtracting, and Multiplying
Radical Expressions
1. Add or subtract like radicals.
2. Use the distributive property in expressions containing
radicals.
3. Simplify radical expressions that contain mixed
operations.
Copyright © 2011 Pearson Education, Inc.
Like radicals: Radical expressions with identical
radicands and identical root indices.
Adding Like Radicals
To add or subtract like radicals, add or subtract the
coefficients and leave the radical parts the same.
Copyright © 2011 Pearson Education, Inc.
Slide 10- 79
Example 1
Add or subtract.
a. 6 7  3 7
b. 4 3 5  2 3 5
Solution
a.
6 7  3 7  (6  3) 7
9 7
b. 4 3 5  2 3 5  (4  2) 3 5
 63 5
Copyright © 2011 Pearson Education, Inc.
Slide 10- 80
continued
Simplify.
c. 7 3 5x  12 3 5x
d. 14 5  11  2 11  5 5
Solution
c. 7 3 5x  12 3 5x  5 3 5x
d. 14 5  11  2 11  5 5
Combine the like radicals by
subtracting the coefficients and
keeping the radical.
Regroup the terms.
 (14  5) 5  (2  1) 11
 9 5  11
Copyright © 2011 Pearson Education, Inc.
Slide 10- 81
Example 2
Add or subtract.
a. 28  7
b. 4 3 135  2 3 5
Solution
a.
28  7  4  7  7
2 7 7
3 7
Factor 28.
Simplify.
Combine like radicals.
b. 4 3 135  2 3 5  4 3 27  5  2 3 5
 4  33 5  2 3 5
 12 3 5  2 3 5  10 3 5
Copyright © 2011 Pearson Education, Inc.
Slide 10- 82
continued
c. 63x5  112x5  28x5
Solution
c.
4
4
4
63x5  112x5  28x5  9 x  7 x  16 x  7 x  4 x  7 x
 3x 2 7 x  4 x 2 7 x  2 x 2 7 x
 3x 2 7 x
Copyright © 2011 Pearson Education, Inc.
Slide 10- 83
Example 3a
Find the product. 3 6
Solution
3 6

5 7 7


5 7 7

 3 6  5  3 6 7 7
 3 30  21 42
Copyright © 2011 Pearson Education, Inc.
Use the distributive
property.
Multiply.
Slide 10- 84
Example 3c

Find the product. 4 5  2
Solution
4
5 2

5 5 2


5 5 2 .

 4 55  5 4 5 2  5 2  5 2 2
Use the distributive
property.
 4  5  20 10  10  5  2
Use the product rule.
 20  20 10  10  10
Find the products.
 10  19 10
Combine like radicals.
Copyright © 2011 Pearson Education, Inc.
Slide 10- 85
Example 3d


Find the product. 4 x  y 3 x  7 y
Solution
4

x  y 3 x 7 y


 4 x 3 x  4 x 7 y  y 3 x  y 7 y
 12x  28 xy  3 xy  7 y
 12 x  25 xy  7 y
Copyright © 2011 Pearson Education, Inc.
Slide 10- 86
Example 3e
Find the product.

5 3

2
Solution

  5
2
5 3 
2
 2 53 
 5  2 15  3
 3
2
Use (a – b)2 = a2 – 2ab – b2.
Simplify.
 8  2 15
Copyright © 2011 Pearson Education, Inc.
Slide 10- 87
Example 4a


Find the product. 8  3 8  3

Solution
8  3 8  3 
8 
2
 3
 64  3
2
Use (a + b)(a – b) = a2 – b2.
Simplify.
 61
Copyright © 2011 Pearson Education, Inc.
Slide 10- 88
Example 4b
Find the product.
Solution

7 2 3


7 2 3

7 2 3

  7   2 3
7 2 3 
2
2
 7  43
 7 12
 5
Copyright © 2011 Pearson Education, Inc.
Slide 10- 89
Example 5
Simplify.
a. 2  14  3  21
Solution
2  14  3  21
a.
 2 14  3  21
 28  63
b.
b.
96
 50
3
96
 50  32  25  2
3
 16  2  5 2
 16  2  5 2
 47  97
 2 7 3 7
 4 2 5 2
9 2
5 7
Copyright © 2011 Pearson Education, Inc.
Slide 10- 90
Simplify. 2 12  5 27  48  2 3
a) 9  3
b) 17 3
c) 71 3
d) 25 3
Copyright © 2011 Pearson Education, Inc.
Slide 10- 91
Simplify. 2 12  5 27  48  2 3
a) 9  3
b) 17 3
c) 71 3
d) 25 3
Copyright © 2011 Pearson Education, Inc.
Slide 10- 92
Multiply.  3 2  3 2  5 3 
a) 9  15 6  3 2
b) 6  3 2  15 3  15 6
c) 5  14 6  15 3
d) 25 3
Copyright © 2011 Pearson Education, Inc.
Slide 10- 93
Multiply.  3 2  3 2  5 3 
a) 9  15 6  3 2
b) 6  3 2  15 3  15 6
c) 5  14 6  15 3
d) 25 3
Copyright © 2011 Pearson Education, Inc.
Slide 10- 94
10.5
Rationalizing Numerators and
Denominators of Radical Expressions
1. Rationalize denominators.
2. Rationalize denominators that have a sum or difference
with a square root term.
3. Rationalize numerators.
Copyright © 2011 Pearson Education, Inc.
Example 1a
Rationalize the denominator.
Solution
8
5
8

5
5
5
8 5

25
8
5
Multiply by
5
.
5
Simplify.
8 5

5
Copyright © 2011 Pearson Education, Inc.
Slide 10- 96
Example 1b
Rationalize the denominator.
Solution
Use the quotient rule for
square roots to separate the
numerator and denominator
into two radicals.
2
2

3
3
2

3
2
3
3
3
Multiply by
6

9
6

3
3
.
3
Simplify.
Warning: Never divide out factors common
to a radicand and a number not under a
radical.
Copyright © 2011 Pearson Education, Inc.
Slide 10- 97
Example 1c
Rationalize the denominator.
5
3x
Solution
5
5
3x


3x
3x 3x

5 3x
9x2
5 3x

3x
Copyright © 2011 Pearson Education, Inc.
Slide 10- 98
Rationalizing Denominators
To rationalize a denominator containing a single nth
root, multiply the fraction by a well chosen 1 so that
the product’s denominator has a radicand that is a
perfect nth power.
Copyright © 2011 Pearson Education, Inc.
Slide 10- 99
Example 2a
Rationalize the denominator. Assume that variables
represent positive values. 5
3
Solution
3
5
5 39
 3 3
3
3
3 9
53 9
3
27
53 9

3
Copyright © 2011 Pearson Education, Inc.
Slide 10- 100
Example 2b
Rationalize the denominator. Assume that variables
3
represent positive values. w
3
Solution
z
w 3 w 3 z2
 3 
3
z 3 z2
z
3
wz 2

3 3
z
3
3
wz 2

z
Copyright © 2011 Pearson Education, Inc.
Slide 10- 101
Example 2c
Rationalize the denominator. Assume that variables
7
represent positive values.
3
16x 2
Solution
3
7
7

3
3
16x 2
16x 2
3
7

3
4x

3
3
4x
16 x 2
Copyright © 2011 Pearson Education, Inc.
3
3
28 x
64 x 3
3
28 x

4x
Slide 10- 102
Rationalizing a Denominator Containing a Sum or
Difference
To rationalize a denominator containing a sum or
difference with at least one square root term, multiply
the fraction by a 1 whose numerator and denominator
are the conjugate of the denominator.
Copyright © 2011 Pearson Education, Inc.
Slide 10- 103
Example 3a
Rationalize the denominator and simplify. Assume
7
variables represent positive values.
Solution
3 5
7
7
3 5


3 5 3 5
3 5
7( 3  5)

( 3) 2  (5) 2
1(7 3  35)

1(22)
7 3  35

3  25
7 3  35

22
35  7 3

22
Copyright © 2011 Pearson Education, Inc.
Slide 10- 104
Example 3b
Rationalize the denominator and simplify. Assume
12 5
variables represent positive values.
Solution
11  3
12 5
11  3
12 5


11  3 11  3
11  3
12 5 11  3

4(3 55  3 15)
( 11)2  ( 3) 2

8
12 55  12 15

11  3
3 55  3 15

12 55  12 15
2

8
Slide 10- 105

Copyright © 2011 Pearson Education, Inc.

Example 3c
Rationalize the denominator and simplify. Assume
4
variables represent positive values.
x 3
Solution
4

x 3

4
x 3

x 3 x 3
4 x 3

 x

2
  3
2
4 x  12

x 9
Copyright © 2011 Pearson Education, Inc.
Slide 10- 106
Example 4a
Rationalize the numerator. Assume variables
3x
represent positive values.
8
Solution
3x
8
3x 3x


8
3x


9 x2

8 3x
3x

8 3x
Copyright © 2011 Pearson Education, Inc.
Slide 10- 107
Example 4b
Rationalize the numerator. Assume variables
represent positive values. 5  3 x
6
Solution
5  3x
6
5  3x 5  3x


6
5  3x
5 
2



6 5

3x 
2
3x
25  3x

30  6 3x
Copyright © 2011 Pearson Education, Inc.
Slide 10- 108
Rationalize the denominator.
8
3
a) 2 3
3
3 2
b)
3
c) 2 6
3
3 6
d)
3
Copyright © 2011 Pearson Education, Inc.
Slide 10- 109
Rationalize the denominator. 8
3
a) 2 3
3
3 2
b)
3
c) 2 6
3
3 6
d)
3
Copyright © 2011 Pearson Education, Inc.
Slide 10- 110
Rationalize the denominator.
5
13  7
a) 5 13  5 7
6
b) 5 13  5 7
c)
6
13  7
5
d)
13  7
5
Copyright © 2011 Pearson Education, Inc.
Slide 10- 111
Rationalize the denominator.
5
13  7
a) 5 13  5 7
6
b) 5 13  5 7
c)
6
13  7
5
d)
13  7
5
Copyright © 2011 Pearson Education, Inc.
Slide 10- 112
10.6
Radical Equations and Problem Solving
1. Use the power rule to solve radical equations.
Copyright © 2011 Pearson Education, Inc.
Radical equation: An equation containing at least one
radical expression whose radicand has a variable.
Power Rule for Solving Equations
If both sides of an equation are raised to the same
integer power, the resulting equation contains all
solutions of the original equation and perhaps some
solutions that do not solve the original equation. That
is, the solutions of the equation a = b are contained
among the solutions of an = bn, where n is an integer.
Copyright © 2011 Pearson Education, Inc.
Slide 10- 114
Example 1
Solve.
a. y  12
Solution
2
a.
y  122
 
b.
b.
3
x  4
 x
3
3
x  64
y  144
Check
  4 
3
Check
144  12
12  12
3
64  4
4   4
True
True
Copyright © 2011 Pearson Education, Inc.
Slide 10- 115
Example 2a
Solve.
Solution
x 5  6
Check:
x 5  6

x 5

2
x 5  6
41  5  6
 (6)2
36  6
x  5  36
66
x  41
The number 41
checks. The solution
is 41.
Copyright © 2011 Pearson Education, Inc.
Slide 10- 116
Example 2b
Solve.
3
x  4  2
Solution
3

3
Check:
x  4  2
x4

3
3
 (2)3
x  4  8
x  4  2
3
4  4  2
3
8  2
2   2
x  4
True. The solution
is 4.
Copyright © 2011 Pearson Education, Inc.
Slide 10- 117
Example 2c
Solve. 4 x  1  5
Solution
Check:
4 x  1  5


4 x  1  5
2
4 x  1  (5) 2
4x  1  25
4x  24
x6
Copyright © 2011 Pearson Education, Inc.
4(6)  1  5
25  5
5  5
False, so 6 is
extraneous. This
equation has no real
number solution.
Slide 10- 118
Example 3a
Solve. 4 x  x  60
Solution
Check:
4 x  x  60
4 x   
2
4
2
 x
2
x  60
4 x  x  60

2
4 4  4  60
4  2  64
 x  60
88
16x  x  60
15x  60
x4
The number 4 checks.
The solution is 4.
Copyright © 2011 Pearson Education, Inc.
Slide 10- 119
Example 4
Solve. x  5  x  7
Solution
x 5  x 7
 x  5
2


x7

2
x 2  10 x  25  x  7
x 2  11x  25  7
x 2  11x  18  0
( x  2)( x  9)  0
x2  0
or x  9  0
x2
x 9
Copyright © 2011 Pearson Education, Inc.
Square both sides.
Use FOIL.
Subtract x from both sides.
Subtract 7 from both sides.
Factor.
Use the zero-factor theorem.
Slide 10- 120
x 5  x 7
continued
Checks
x2
x 9
95  97
25  27
4  16
3  9
3  3
False.
44
True.
Because 2 does not check, it is an extraneous
solution. The only solution is 9.
Copyright © 2011 Pearson Education, Inc.
Slide 10- 121
Example 5a
Solve. x  4  6
Solution
x  4  6
Check
4  4  6
x  2
 x
2
  2 
x  4  6
2  4  6
2  6
2
x4
This solution does not check, so it is an
extraneous solution. The equation has no real
number solution.
Copyright © 2011 Pearson Education, Inc.
Slide 10- 122
Example 5b
Solve
4
x  3  3  5.
Solution
4
x3 3 5
4

4
x3  2
x3

4
4
Check
4
x3 3 5
13  3  3  5
 24
4
16  3  5
x  3  16
23  5
x  13
55
The solution set is 13.
Copyright © 2011 Pearson Education, Inc.
Slide 10- 123
Example 6
x  16  x  8
Solve
Solution
Check
x  16  x  8

x  16
 
2
x 8

2
x  16  x  8 x  8 x  64
x  16  x  16 x  64
48  16 x
x  16  x  8
9  16  9  8
25  3  8
5  11
There is no solution.
3  x
(3) 2  ( x ) 2
9 x
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Slide 10- 124
Solving Radical Equations
To solve a radical equation,
1. Isolate the radical if necessary. (If there is more
than one radical term, isolate one of the radical
terms.)
2. Raise both sides of the equation to the same
power as the root index of the isolated radical.
3. If all radicals have been eliminated, solve. If
a radical term remains, isolate that radical
term and raise both sides to the same power as its
root index.
4. Check each solution. Any apparent solution that
does not check is an extraneous solution.
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Slide 10- 125
Solve.
5x  4  x  2
a) 6
b) 8
c) 9
d) no solution
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Slide 10- 126
Solve.
5x  4  x  2
a) 6
b) 8
c) 9
d) no solution
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Slide 10- 127
Solve. 4a  10  2  a
a) 2
b) 4
12
c)
5
d) no real-number solution
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Slide 10- 128
Solve. 4a  10  2  a
a) 2
b) 4
12
c)
5
d) no real-number solution
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Slide 10- 129
Solve. x  2  5x  16
a) 3, 4
b) 3
c) 4
d) no real-number solution
Copyright © 2011 Pearson Education, Inc.
Slide 10- 130
Solve. x  2  5x  16
a) 3, 4
b) 3
c) 4
d) no real-number solution
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Slide 10- 131
10.7
Complex Numbers
1. Write imaginary numbers using i.
2. Perform arithmetic operations with complex numbers.
3. Raise i to powers.
Copyright © 2011 Pearson Education, Inc.
Imaginary unit: The number represented by i, where
i  1 and i2 = 1.
Imaginary number: A number that can be expressed
in the form bi, where b is a real number and i is the
imaginary unit.
Copyright © 2011 Pearson Education, Inc.
Slide 10- 133
Example 1
Write each imaginary number as a product of a real
number and i.
a. 16
b. 21
c. 32
Solution
a. 16
b.
21
c.
32
 116
 1 21
 1 32
 1  16
 i4
 4i
 1  21
 1  32
 i 21
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 i 16  2
 4i 2
Slide 10- 134
Rewriting Imaginary Numbers
To write an imaginary number  n in terms of the
imaginary unit i,
1. Separate the radical into two factors, 1  n .
2. Replace 1 with i.
3. Simplify n .
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Slide 10- 135
Complex number: A number that can be expressed in
the form a + bi, where a and b are real numbers and i
is the imaginary unit.
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Slide 10- 136
Example 2a
Add or subtract. (9 + 6i) + (6 – 13i)
Solution
We add complex numbers just like we add
polynomials—by combining like terms.
(9 + 6i) + (6 – 13i) = (9 + 6) + (6i – 13i )
= –3 – 7i
Copyright © 2011 Pearson Education, Inc.
Slide 10- 137
Example 2b
Add or subtract. (3 + 4i) – (4 – 12i)
Solution
We subtract complex numbers just like we subtract
polynomials.
(3 + 4i) – (4 – 12i) = (3 + 4i) + (4 + 12i)
= 7 + 16i
Copyright © 2011 Pearson Education, Inc.
Slide 10- 138
Example 3
Multiply.
a. (8i)(4i)
b. (6i)(3 – 2i)
Solution
a. (8i)(4i)  32i 2
b. (6i)(3 – 2i)
2

18
i

12
i
 32(1)
 18i  12(1)
 32
 18i 12
 12 18i
Copyright © 2011 Pearson Education, Inc.
Slide 10- 139
continued
Multiply.
c. (9 – 4i)(3 + i)
d. (7 – 2i)(7 + 2i)
Solution
c. (9 – 4i)(3 + i)
 27  9i  12i  4i 2
d. (7 – 2i)(7 + 2i)
 49  14i  14i  4i 2
 27  3i  4(1)
 49  4( 1)
 27  3i  4
 49  4
 31  3i
 53
Copyright © 2011 Pearson Education, Inc.
Slide 10- 140
Complex conjugate: The complex conjugate of a
complex number a + bi is a – bi.
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Slide 10- 141
Example 4a
7
Divide. Write in standard form. 3i
Solution Rationalize the denominator.
7
7 i
7i
 

3i
3i i
3
7i
 2
3i
7i

3(1)
7i

3
Copyright © 2011 Pearson Education, Inc.
Slide 10- 142
Example 4b
3  5i
Divide. Write in standard form.
5i
Solution Rationalize the denominator.
15  3i  25i  5
3  5i 3  5i 5  i



25  1
5i
5i 5i
10  28i
2

15  3i  25i  5i

26
2
25  i
10 28i
15  3i  25i  5(1)



26 26
25  (1)
5 14i
 
13 13
Copyright © 2011 Pearson Education, Inc.
Slide 10- 143
Example 5
40
Simplify. a. i
b. i33
Solution
a. i
40
= i

4 10
= 110 = 1
Write i40 as (i4)10.
b. i 33 = i 32  i
= i

4 8
= 1 i
i
Write i32 as (i4)8.
Replace i4 with 1.
=i
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Slide 10- 144
Simplify. (4 + 7i) – (2 + i)
a) 2 + 7i2
b) 2 + 8i
c) 6 + 6i
d) 6 + 8i
Copyright © 2011 Pearson Education, Inc.
Slide 10- 145
Simplify. (4 + 7i) – (2 + i)
a) 2 + 7i2
b) 2 + 8i
c) 6 + 6i
d) 6 + 8i
Copyright © 2011 Pearson Education, Inc.
Slide 10- 146
Multiply. (4 + 7i)(2 + i)
a) 15 + 10i
b) 1 + 10i
c) 15 + 18i
d) 15 + 18i
Copyright © 2011 Pearson Education, Inc.
Slide 10- 147
Multiply. (4 + 7i)(2 + i)
a) 15 + 10i
b) 1 + 10i
c) 15 + 18i
d) 15 + 18i
Copyright © 2011 Pearson Education, Inc.
Slide 10- 148
Write in standard form. 4  i
2  3i
a)
5 14i

13 13
5 14i

b)
13 13
c)
11 14i

13 13
11 14i
d)

13 13
Copyright © 2011 Pearson Education, Inc.
Slide 10- 149
Write in standard form. 4  i
2  3i
a)
5 14i

13 13
5 14i

b)
13 13
c)
11 14i

13 13
11 14i
d)

13 13
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Slide 10- 150