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Section P3Radicals and Rational Exponents
Square Roots
81 9
40 9 7
9 3
64 8
2
Definition of the Principal Square Root
If a is a nonnegative real number, the nonnegative number b
such that b =a, denoted by b= a is the principal square root of a.
Examples
36 16
100 44
121
Evaluate
Simplifying Expressions
of the Form 2a
The Product Rule for Square Roots
A square root is simplified when its radicand has no factors other than 1 that are perfect squares.
Examples
4900
Simplify:
Examples
4 63x x
Simplify:
The Quotient Rule for Square Roots
Examples
Simplify:
3
9
49
54
2
x
x
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.
Example
10 5 2 5
3 6 3 12
Add or Subtract as indicated:
Example
7 98 2 5 28x x x x
Add or Subtract as indicated:
Rationalizing Denominators
Rationalizing a denominator involves rewriting a radical expression as an equivalent expression in which the denominator no longer contains any radicals. If the denominator contains 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.
Let’s take a look two more examples:
Examples
7
6
7
18
Rationalize the denominator:
Examples
2
3 2 5
Rationalize the denominator:
Other Kinds of Roots
Examples
3
3
4
8
8
16
Simplify:
The Product and Quotient Rules for nth Roots
Example
4
5 5
6 81
4 40
Simplify:
Example
3
3 3
64
27
250 2 16
Simplify:
Rational Exponents
Example
3
4
3
5
5
3
1
2
81
32
48
3
x
x
Simplify:
Example
54 1
5 3
24
2
81
x x
x
Simplify:
Notice that the index reduces on this last problem.
(a)
(b)
(c)
(d)
381
4
x
x
Simplify:
9
29
29
2
9
2
x
x x
x
x
x
(a)
(b)
(c)
(d)
23 1
4 27 3x x
Simplify:
5
4
5
4
7
4
7
4
21
63
21
63
x
x
x
x
