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Section P3 Radicals and Rational Exponents

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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