Simplifying Radicals

Section 5.3

Radicals Definition Simplifying Adding/Subtracting Multiplying Dividing Rationalizing the denominator

Radicals - definitions

The definition of is the number that when multiplied by itself 2 times is x.

x

2224 xxxx 2

16

Simplifying radicals

Most numbers are not perfect squares, but may have a factor(s) that is (are) a perfect square(s).

The perfect squares are: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, ….

Try these - simplify:

If a radical has a perfect square factor, then we can pull it out from under the sign.

Ex:

98724832

50 225 25 25

745 bax

3 3 727m n

Adding or Subtracting Radicals

24232

24232182

To add or subtract square roots you must have like radicands (the number under the radical).

Sometimes you must simplify first:

Try These

5553 5553

48753 803202

18554

Multiplying Radicals

62343232

10152553

572 10253 2

2

You can multiply any square roots together. Multiply any whole numbers together and then multiply the numbers under the radical and reduce.

Try these:

)32)(123( 2

52

Dividing Radicals

To divide square roots, divide any whole numbers and then divide the radicals one of two ways:

1) divide the numbers under the radical sign and then take the root, OR

2) take the root and then divide. Be sure to simplify.

5

20

5

20

25

52

5

20 24

5

20 or

Try These

4

1

10

40

4

100

22

148

25

100

10

80

Rationalizing Radicals

It is good practice to eliminate radicals from the denominator of an expression.

For example:

We do not want to change the value of the expression, so we need to multiply the fraction by 1. But “1” can be written in many ways…

2

3 2

222 2

21

2

23

22

23

2

2

2

3

We need to eliminate

Since we will multiply by one where

Try These

3

5

5

22

10

53

12

35

83

52

20

7