Copyright © Cengage Learning. All rights reserved. Roots, Radical Expressions, and Radical Equations 8

Copyright © Cengage Learning. All rights reserved. Section 8.3 Simplifying Radical Expressions

3 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

4 Simplify a radical expression using the multiplication property of radicals 1.

5 Simplify a radical expression using the multiplication property of radicals 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,.

6 Simplify a radical expression using the multiplication property of radicals 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.

7 Simplify a radical expression using the multiplication property of radicals 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.

8 Simplify a radical expression using the multiplication property of radicals 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.

9 Simplify a radical expression using the multiplication property of radicals To simplify more difficult radicals, we need to know the integers that are perfect squares. For example, 81 is a perfect square, because 9 2 = 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, 9x 4 y 2 is a perfect square, because 9x 4 y 2 = (3x 2 y) 2

10 Example Simplify: (x  0). Solution: We factor 72x 3 into two factors, one of which is the greatest perfect square that divides 72x 3. Since 36 is the greatest perfect square that divides 72, and x 2 is the greatest perfect square that divides x 3, the greatest perfect square that divides 72x 3 is 36x 2.

11 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

12 Simplify a radical expression using the division property of radicals 2.

13 Simplify a radical expression using the division property of radicals 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,.

14 Simplify a radical expression using the division property of radicals 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.

15 Simplify a radical expression using the division property of radicals We can use the division property of radicals to simplify radicals that have fractions in their radicands. For example,

16 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.

17 Simplify a cube root expression 3.

18 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, 27x 6 y 3 is a perfect cube, because 27x 6 y 3 = (3x 2 y) 3

19 Example Simplify: a. b. (m  0). Solution: a. We look for the greatest perfect cube that divides 16x 3 y 4. Because 8 is the greatest perfect cube that divides 16, x 3 is the greatest perfect cube that divides x 3, and y 3 is the greatest perfect cube that divides y 4, the greatest perfect-cube factor that divides 16x 3 y 4 is 8x 3 y 3.

20 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

21 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

22 Simplify a cube root expression 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.

23 Simplify a cube root expression Remember that it is incorrect to write