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

2. Copyright © Cengage Learning. All rights reserved. Section 8.5 Multiplying and Dividing Radical Expressions

3. 3 Objectives Multiply two or more radical expressions. Divide two single-term radical expressions. Rationalize an expression with a single-term denominator. Rationalize an expression with a denominator having two terms. 1 1 2 2 3 3 4 4

4. 4 Multiplying and Dividing Radical Expressions We will now discuss how to multiply and divide radical expressions. The process for multiplication is similar to the process of multiplying polynomials.

5. 5 Multiply two or more radical expressions 1.

6. 6 Multiply two or more radical expressions Recall that the product of the square roots of two nonnegative numbers is equal to the square root of the product of those numbers. For example,

7. 7 Multiply two or more radical expressions Likewise, the product of the cube roots of two numbers is equal to the cube root of the product of those numbers. For example, To multiply radical expressions, we multiply the coefficients and multiply the radicals separately and then simplify the result, when possible.

8. 8 Example Multiply: a. b. (x  0) Solution: The commutative and associative properties enable us to multiply the integers and the radicals separately. a.

9. 9 Example – Solution b. cont’d

10. 10 Multiply two or more radical expressions Recall that to multiply a polynomial by a monomial, we use the distributive property. We can use the same process to multiply radical expressions. To multiply two binomials, we multiply each term of one binomial by each term of the other binomial and simplify. We can use this same process to multiply radical expressions, each containing two terms.

11. 11 Divide two single-term radical expressions 2.

12. 12 Divide two single-term radical expressions To divide radical expressions, we use the division property of radicals. For example, to divide by, we proceed as follows: The quotient of two square roots is the square root of the quotient.

13. 13 Example Divide: (a  0, b  0). Solution: We can write the quotient of two radicals as the radical of a quotient and then simplify. Simplify the radicand.

14. 14 Example – Solution The square root of a quotient is equal to the quotient of the square roots. The square root of a product is equal to the product of the square roots. Simplify the radicals. cont’d

15. 15 Rationalize an expression with a single-term denominator 3.

16. 16 Rationalize an expression with a single-term denominator The fraction is not written in simplest form because a radical appears in its denominator. We can use a process called rationalizing the denominator to simplify such fractions. For example, to eliminate the radical in the denominator of, we can multiply the fraction by 1, written in the form. This will result in a denominator that is the rational number 2, because.

17. 17 Rationalize an expression with a single-term denominator Comment When rationalizing a denominator, always multiply by a value that will result in the removal of the radical sign in the denominator. Multiply the fraction by 1:.

18. 18 Example Rationalize each denominator: a. b.. Solution: a. We multiply the numerator and denominator by because and 3 is a rational number. Then we simplify. Multiply the fraction by 1:.

19. 19 Example – Solution b. We need to multiply by a value that will produce a perfect cube under the radical. Since 27 is a perfect-integer cube, we multiply the numerator and denominator by and simplify. cont’d

20. 20 Rationalize an expression with a denominator having two terms 4.

21. 21 Rationalize an expression with a denominator having two terms Since the denominators of many fractions such as contain radicals, they are not in simplest form. Because the denominator has two terms, multiplying the denominator by will not make it a rational number. The key to rationalizing this denominator is to multiply the numerator and denominator by because the product has no radicals. Radical expressions such as and are called conjugates of each other.

22. 22 Example Divide: Solution: Multiply the numerator and denominator by the conjugate of the denominator. Multiply the binomials in the denominator. Multiply numerator and denominator by the conjugate of the denominator:

23. 23 Example – Solution Simplify. Divide out the common factor of 2. cont’d