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