10-1A Simplifying Radicals

Algebra 1 Glencoe McGraw-Hill Linda Stamper

105

You have simplified fractions.

answer

21

Simplifying Radicals

The simplest form of a radical expression is an expression that has:

No perfect square factors other than 1 in the radicand.

20

259

not simplified

No fractions in the radicand.

not simplified

No radicals in the denominator of a fraction. 2

1not simplified

Use the properties of radicals to simplify radical expressions.All radical

answers must be in simplified

form!

Product Property of Radicals

.0b and 0a where baab

20

54

52

Rewrite using a perfect square factor.

48316

54

316

34

48

34316344

124

An efficient method is to find the

largest perfect square factor.

Write each factor as a radical.

Simplify.

Simplify.

98

Example 1

Example 2

Example 363 40

180

Example 4

Example 5

Example 6150 52

Simplify.

98 249

249

27

Why isn’t the answer Because the problem was not

? 27

. 98

Example 1

Example 2

Example 363 79

79

73

40 104

104

102

Simplify.

180 209

549

56

Example 4

Example 5

Example 6150 625

625

65

52 134

134

132

549

523

Product Property of Radicals

.0b and 0a where baab

20

54

52

Rewrite using a perfect square factor.

54

Write each factor as a radical.

Simplify.

Multiply radicals using the product property.

.0b and 0a where ab ba 322 322

648

Rewrite as one radical.Simplify.

Simplify.

153 153

53

Example 7

Example 8

Example 986 86

48

34

63104 61034 6012

1524

4559 316 15412

15212

Simplify a square root with variables.

2x x

When finding the principal square root of an expression containing variables, be sure that the result is not negative.

22 2It may seem that the answer is…What if x has a value of -2.

22 Substitute -2 for x in the equation.

?

For radical expressions where the exponent of the variable inside the radical is eveneven and the resulting simplified exponent is oddodd, you must use absolute value to ensure nonnegative results.

2x x 4x2x

3xxx

12 xx 5x

xx2

14 xx 22 xx

4 2?

zzyyyxx52 222222

Simplify.

354 zyx20

Write the radicand as prime factors.

Write the problem.

52 x x ySimplify.

y y z z

Use good form – alphabetical order (inside

and outside of the radical) with

radical last.

zzyyx52 22

yz5zyx2 22

If the power of the variable is

an odd number, write the

variable with absolute value

bars

222 yx63

Simplify.

22yx54You can have more than one variable in absolute value bars.

6xy3

Simplify.

24yx81

Example 10

Example 11

Example 1224ba24 yx100 6

Simplify.

24yx81

Example 10

Example 11

Example 1224ba24

2222 baa62

yx100 6

yxxx10 2222

yx9 2 ba62 2

6ba2 2

yx10 3

2222 yxx9

10-A2 Pages 532-534 #17–28,68-70.