section p3 radicals and rational exponents. square roots
TRANSCRIPT
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Section P3Radicals and Rational Exponents
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Square Roots
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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.
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Examples
36 16
100 44
121
Evaluate
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Simplifying Expressions
of the Form 2a
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The Product Rule for Square Roots
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A square root is simplified when its radicand has no factors other than 1 that are perfect squares.
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Examples
4900
Simplify:
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Examples
4 63x x
Simplify:
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The Quotient Rule for Square Roots
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Examples
Simplify:
3
9
49
54
2
x
x
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Adding and Subtracting Square Roots
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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
10 5 2 5
3 6 3 12
Add or Subtract as indicated:
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Example
7 98 2 5 28x x x x
Add or Subtract as indicated:
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Rationalizing Denominators
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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.
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Let’s take a look two more examples:
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Examples
7
6
7
18
Rationalize the denominator:
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Examples
2
3 2 5
Rationalize the denominator:
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Other Kinds of Roots
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Examples
3
3
4
8
8
16
Simplify:
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The Product and Quotient Rules for nth Roots
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Example
4
5 5
6 81
4 40
Simplify:
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Example
3
3 3
64
27
250 2 16
Simplify:
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Rational Exponents
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Example
3
4
3
5
5
3
1
2
81
32
48
3
x
x
Simplify:
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Example
54 1
5 3
24
2
81
x x
x
Simplify:
Notice that the index reduces on this last problem.
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(a)
(b)
(c)
(d)
381
4
x
x
Simplify:
9
29
29
2
9
2
x
x x
x
x
x
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(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