copyright © cengage learning. all rights reserved. roots, radical expressions, and radical...

Copyright © Cengage Learning. All rights reserved.

Roots, Radical Expressions,and Radical Equations 8

Copyright © Cengage Learning. All rights reserved.

Section 8.38.3

Simplifying Radical Expressions

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Objectives

Simplify a radical expression using the multiplication property of radicals.

Simplify a radical expression using the division property of radicals.

Simplify a cube root expression.

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33

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Simplify a radical expression using the multiplication property of radicals

1.

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We introduce the first of two properties of radicals with the following examples:

In each case, the answer is 10. Thus, .

Likewise,

In each case, the answer is 12. Thus, .

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These results suggest the multiplication property of radicals.

Multiplication Property of Radicals

If a 0 and b 0, then

In words, the square root of the product of two nonnegative numbers is equal to the product of their square roots.

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A square-root radical is in simplified form when each of the following statements is true.

Simplified Form of a Square Root Radical

1. Except for 1, the radicand has no perfect-square factors.

2. No fraction appears in a radicand.

3. No radical appears in the denominator of a fraction.

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We can use the multiplication property of radicals to simplify radicals that have perfect-square factors.

For example, we can simplify as follows:

Factor 12 as 4 3, because 4 is a perfect square.

Use the multiplication property of radicals:

Simplify.

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To simplify more difficult radicals, we need to know the integers that are perfect squares.

For example, 81 is a perfect square, because 92 = 81. The first 20 integer squares are

1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400

Expressions with variables also can be perfect squares. For example, 9x4y2 is a perfect square, because

9x4y2 = (3x2y)2

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Example

Simplify: (x 0).

Solution:

We factor 72x3 into two factors, one of which is the greatest perfect square that divides 72x3.

Since

• 36 is the greatest perfect square that divides 72, and

• x2 is the greatest perfect square that divides x3,

the greatest perfect square that divides 72x3 is 36x2.

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Example – Solution

We can now use the multiplication property of radicals and simplify to get

The square root of a product is equal to the product of the square roots.

Simplify.

cont’d

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Simplify a radical expression using the division property of radicals

2.

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To find the second property of radicals, we consider these examples.

and

= 2 = 2

Since the answer is 2 in each case, .

Likewise,

= 3 = 3

Since the answer is 3 in each case, .

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These results suggest the division property of radicals.

Division Property of Radicals

If a 0 and b > 0, then

In words, the square root of the quotient of a nonnegative number and a positive number is the quotient of their square roots.

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We can use the division property of radicals to simplify radicals that have fractions in their radicands. For example,

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Example

Simplify: .

Solution:

The square root of a quotient is equal to the quotient of the square roots.

Factor 108 using the factorization involving 36, the largest perfect-square factor of 108, and write as 5.

The square root of a product is equal to the product of the square roots.

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Simplify a cube root expression3.

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Simplify a cube root expression

The multiplication and division properties of radicals are also true for cube roots and higher.

To simplify a cube root, it is helpful to know the following integer cubes:

1, 8, 27, 64, 125, 216, 343, 512, 729, 1,000

Expressions with variables can also be perfect cubes. For example, 27x6y3 is a perfect cube, because

27x6y3 = (3x2y)3

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Example

Simplify: a. b. (m 0).

Solution:a. We look for the greatest perfect cube that divides 16x3y4.

Because

• 8 is the greatest perfect cube that divides 16,

• x3 is the greatest perfect cube that divides x3, and

• y3 is the greatest perfect cube that divides y4,

the greatest perfect-cube factor that divides 16x3y4 is 8x3y3.

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Example 6 – Solution

We now can use the multiplication property of radicals to obtain

The cube root of a product is equal to the product of the cube roots.

Simplify.

cont’d

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Example – Solution

The cube root of a quotient is equal to the quotient of the cube roots.

Use the multiplication property of

radicals, and write as 3m.

Simplify.

cont’d

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Comment

Note that and .

To see that this is true, we consider these correct simplifications:

and

Since the radical sign is a grouping symbol, the order of operations requires that we perform the operations under the radicals first.

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Remember that it is incorrect to write

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