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Chapter P
Prerequisites:
Fundamental
Concepts of Algebra
P.3 Radicals and
Rational Exponents
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Objectives:
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Evaluate square roots.
Simplify expressions of the form a 2
Use the product rule to simplify square roots.
Use the quotient rule to simplify square roots.
Add and subtract square roots.
Rationalize denominators.
Evaluate and perform operations with higher roots.
Understand and use rational exponents.
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Definition of the Principal Square Root.
In general, if b2 = a, then b is the square root of a.
Definition of the Principal Square Root:
If a is a non-negative real number, the nonnegative
number b such that b2 = a denoted by b  a , is the
principal square root of a.
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Example: Evaluating Square Roots
Evaluate:
81  9
Evaluate:  9  3
Evaluate:
36  64  100  10
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Simplifying Expressions of the Form
Simplifying
a
a2
2
For any real number a,
a2  a
In words, the principal square root of a2 is the absolute
value of a.
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Example: Simplifying Expressions of the Form
a2
Simplify:
52  5
Simplify:
(5)2  25  5
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The Product Rule for Square Roots
If a and b represent nonnegative real numbers, then
ab  a  b
and
a  b  ab
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Example: Using the Product Rule to Simplify Square
Roots
Simplify:
75  25  3  25 3  5 3
Simplify:
5 x  10 x  50x 2  25 x 2  2  25 x 2 2
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 5x 2
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The Quotient Rule for Square Roots
If a and b represent nonnegative real numbers and b  0 ,
then
a
a

b
b
and
a
a

b
b
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Example: Using the Quotient Rule to Simplify Square
Roots
Simplify:
25
16
25

16
5

4
Simplify:
150 x3
150 x 3

 75x 2  25 x 2 3  5 x 3
2x
2x
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Adding and Subtracting Square Roots
Two or more square roots can be combined using the
distributive property provided that they have the same
radicand. Such radicals are called like radicals.
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Example: Adding and Subtracting Like Radicals
Add:
8 13  9 13  (8  9) 13  17 13
Subtract:
17 x  20 17 x  (1  20) 17x  19 17x
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Rationalizing Denominators
The process of rewriting a radical expression as an
equivalent expression in which the denominator no
longer contains any radicals is called rationalizing the
denominator. If the denominator consists of the square
root of a natural number that is not a perfect square,
multiply the numerator and the denominator by the
smallest number that produces the square root of a
perfect square in the denominator.
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Rationalizing Denominators (continued)
Radical expressions that involve the sum and difference
of the same two terms are called conjugates. Thus,
and
a b
a b
are conjugates. If the denominator contains two terms
with one or more square roots, multiply the numerator
and denominator by the conjugate of the
denominator.
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Example: Rationalizing Denominators
Rationalize the denominator:
5
5
3
5 3
5 3




3
3
3 3
9
Rationalize the denominator:
8(4  5)
8
8
4 5



2
2
4 5 4 5 4 5 4  5
 
8(4  5)

11
32  8 5

11
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Definition of the Principal nth Root of a Real Number
n
a  b means that bn = a
If n, the index, is even, then a is nonnegative (a  0)
and b is also nonnegative (b  0) . If n is odd, a and b
can be any real numbers. The symbol n
is called a
radical and the expression under the radical is called
the radicand.
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Finding nth Roots of Perfect nth Powers
If n is odd,
n
aa
If n is even,
n
aa
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The Product and Quotient Rules for nth Roots
For all real numbers a and b, where the indicated roots
represent real numbers,
n
n
ab  n a  n b
a na
n
b
b
and
 b  0  and
n
a  n b  n ab
n
a na

n
b
b
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b  0
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Example: Simplifying, Multiplying, and Dividing Higher
Roots
Simplify:
3
40  3 8  3 5  2 3 5
Simplify:
3
125
5
125
3

 3
27
3
27
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The Definition of a
m
n
m
If n a represents a real number and n is a positive
rational number, n  2 , then
m
n
a 
 a
n
m
m
n
Also,
a  n am
Furthermore, if a

m
n
a
is a nonzero real number, then

m
n

1
a
m
n
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Example: Using the definition of a
m
n
Simplify:
27
4
3


3
27

4
 34  81
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Example: Simplifying Expressions with Rational
Exponents
Simplify using properties of exponents:



 2 x  5 x   2  5  x  x



4
3
8
3
4
3
8
3
 10x
48
3
 10x
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3
 10x 4
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