4

16

25

100

144

= 2

= 4

= 5= 10

= 12

Simplify

Objective - To simplify rational and irrational square roots.

List of Perfect Squares

3649

64

81

100

121

67891011

144

169

196

225

400

625

121314152025

256 16

289 17

324 18

361 19

Objective - To simplify irrational roots.

Example:

Check using a calculator!

Simplify.

To simplify the number by hand must have a perfect square factor.

No perfectsquare factors Can’t be

simplified

1)

2)

3)

4)

5)

6)

Simplify each irrational root.

Simplify.

Perfect square

Simplify each irrational root.

1)

2)

3)

4)

5)

6)80 16 5 16 5 4 5

Already Simplified

Ex. 1 Simplify

40 75 32

450 50 80

Ex. 2 Simplify

To simplify radicals with variables: divide even exponents by 2; for odd exponents, pull out the largest even exponent, take the square root of that, and leave the remainder under the radical.

Radicals with Variables

Radicals with VariablesEx. 10 Even

Radicals with VariablesEx. 11 Odd

+To combine radicals: combine the coefficients of like radicals. They must have the same last name.

Combine Like Radicals

Miller Miller

78

62747365 7763

737576

Ex. 3 Simplify

636556

Your Turn

Simplify, then combine.

485273

Ex. 4

Simplify, then combine.Ex. 5

547243

Slide 10- 17

Copyright © 2007 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Example496 .a b

416 6 a b

416 6a b

24 6 a b

The greatest perfect square factor of 96a4b is 16a4.

Use the product rule of square roots to separate the factors into two radicals.

Find the square root of 16a4 and leave 6b in the radical.

496a b

Slide 10- 18

Example

Solution

532x

416 2x x

24 2 x x

532x The greatest perfect square factor of 32x5 is 16x4.

Use the product rule of square roots to separate the factors into two radicals.

Find the square root of 16x4 and leave 2x in the radical.

To multiply radicals: multiply everything, then simplify.

Multiply Radicals

*

35*5

Ex. 6

204*52

Multiply, then simplify.

2

7

2

8

2

x

Squared and Square Root Cancel

Ex. 7

To divide radicals: reduce or divide what you can and rationalize the denominator. A square root CANNOT be in the denominator.

Dividing Radicals

7

56

7

6

10

5

Ex. 8 Reduce or divide, then

rationalize.

Ex. 9

12

3