Download Chapter 8 Rational Exponents, Radicals, and

Chapter 8
Rational
Exponents,
Radicals, and
Complex
Numbers
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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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8.3
Multiplying, Dividing, and
Simplifying Radicals
1. Multiply radical expressions.
2. Divide radical expressions.
3. Use the product rule to simplify radical
expressions.
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Product Rule for Radicals
If both n a and n b are real numbers, then
n
a  n b  n a  b.
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Example
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
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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
 4 16
d.
3
4 x  3 5x  3 4 x  5x
 3 20x2
2
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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
 y2
9
f.
6
y
6 y



w 5
w 5
6y

5w
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continued
Find the product and simplify. Assume all variables
represent positive values.
g. x2  x2
Solution
g.
x2  x2  x2  x2
 x4
 x2
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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
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Example
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
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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
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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.
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Example
Simplify.
a. 80
b.
Solution
Solution
6 98  6  49  2
80  16 5
 16
4 5
6 98
5
 6  49  2
 67 2
 42 2
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continued
Simplify.
c. 4 3 448
d. 5 4 48
Solution
Solution
4 3 448  4
3
4 4
 16 7
3
3
64
3
7
7
5 4 48  5 4 16  3
 5 4 16  4 3
 5 2 4 3
 10 4 3
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Example
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.
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continued
Simplify the radical using prime factorization.
b. 3 500
c. 4 810
Solution
3
500

2 2555
b.
3
 53 2  2
 53 4
c.
4
810  4 2  3  3  3  3  5
 34 2  5
 3 4 10
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Example
Simplify.
32x5
Solution
32x5  16 x4
 16x4
2x
2x
 4 x2 2 x
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.
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Example
Simplify 2 96a4b.
Solution
2 96a b  2 16a
4
 2 16a
 2  4a
4
4
2
6b
 8a 2 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.
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continued
Simplify.
c. y 3 3 y10
d. 5 486x11 y14
Solution
Solution
y
33
y y
10
33
y
33
y y
y 
9
 y y
3
5
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
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Example
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
Solution
5 8
 40
 4 10
 2 10
4 5x5  5 50 x4
 4  5 5x5  50 x4
 20 250x9
 20 25 10  x8  x
 20  5 x 4 10 x
 100 x 4 10 x
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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
3 5a 5b
Solution
Solution
300
300

4
4
3 5a 5b
245a9b6
3
5a5b
 3 49a4b5
 75
 25  3
5 3
9 245a 9b6
 3 49a4b4  b
 3  7a 2b2 b
 21a 2b2 b
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