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
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
2
8.6
Radical Equations and Problem
Solving
1. Use the power rule to solve radical equations.
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
3
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 © 2015, 2011, 2007 Pearson Education, Inc.
4
Example
Solve.
a. y  12
b.
Solution
2
a.
y  122
 
b.
3
x  4
 x
3
  4 
3
x  64
y  144
Check
3
Check
144  12
12  12
True
3
64  4
4   4
True
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
5
Example
Solve.
Solution
Check:
x 5  6

x 5
x 5  6

2
 (6)2
x  5  36
x  41
x 5  6
41  5  6
36  6
66
The number 41
checks. The solution
is 41.
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
6
Example
Solve.
Solution
3

3
x  4  2
Check:
x  4  2
x4
3

3
 (2)3
x  4  8
3
x  4  2
3
4  4  2
3
8  2
2   2
x  4
True. The solution
is 4.
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
7
Example
Solve.
Solution
Check:
4 x  1  5

4 x  1  5

2
4 x  1  (5) 2
4x  1  25
4x  24
x6
4 x  1  5
4(6)  1  5
25  5
5  5
False, so 6 is
extraneous. This
equation has no real
number solution.
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
8
Example
Solve. 4 x  x  60
Solution
Check:
4 x  x  60
4 x   
2
4
2
 x
2
x  60
 x  60
4 x  x  60

2
4 4  4  60
4  2  64
88
16x  x  60
15x  60
x4
The number 4 checks.
The solution is 4.
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
9
Example
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
Square both sides.
Use FOIL.
Subtract x from both sides.
Subtract 7 from both sides.
Factor.
Use the zero-factor theorem.
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
10
continued x  5  x  7
Checks
x2
25  27
95  97
4  16
3  9
3  3
x 9
False.
44
True.
Because 2 does not check, it is an extraneous
solution. The only solution is 9.
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
11
Example
Solve. x  4  6
Solution
x  4  6
Check
4  4  6
x  2
 x
2
  2 
x4
x  4  6
2
2  4  6
2  6
This solution does not check, so it is an
extraneous solution. The equation has no real
number solution.
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
12
Example
Solve
4
x  3  3  5.
Solution
4
x3 3 5
4

4
x3  2
x3

4
4
Check
4
 24
x3 3 5
13  3  3  5
4
16  3  5
x  3  16
23  5
x  13
55
The solution set is 13.
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
13
Example
x  16  x  8
Solve
Solution
Check
x  16  x  8

x  16
 
2
x 8

2
x  16  x  8
9  16  9  8
x  16  x  8 x  8 x  64
25  3  8
x  16  x  16 x  64
48  16 x
3  x
5  11
There is no solution.
(3) 2  ( x ) 2
9 x
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
14
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.
Copyright © 2015, 2011, 2007 Pearson Education, Inc.
15