Radical Functions Copyright © Cengage Learning. All rights reserved. 8 8.3 Multiplying and Dividing Radicals Copyright © Cengage Learning. All rights reserved. Objectives Multiply radical expressions. Divide radical expressions. 3 Multiplying Radicals 4 Multiplying Radicals To use the product property of radicals, the two radical expressions must have the same index. To multiply two radical expressions with the same index, multiply the radicands (insides) together. After multiplying the radicands, simplify the result if possible. 5 Example 1 – Multiplying radicals Multiply the following and simplify the result. 6 Example 1 – Solution There are no square factors, so it is simplified. 20 has a perfect square factor, so simplify. 7 Example 1 – Solution cont’d 36 and a2 are perfect squares, so simplify. 20 has no perfect cube factors, but the m4 can be simplified. 8 Dividing Radicals and Rationalizing the Denominator 9 Dividing Radicals and Rationalizing the Denominator Division inside a radical can be simplified in the same way that a fraction would be reduced if it were by itself. This follows from the powers of quotients rule for exponents. You can use this rule to simplify some radical expressions that have fractions in them. 10 Dividing Radicals and Rationalizing the Denominator Please note that you can simplify only fractions that are either both inside a radical or both outside the radical. That is, you cannot divide out something that is inside the radical with something that is outside of a radical. 11 Example 4 – Simplifying radicals with division Simplify the following radicals. 12 Example 4 – Solution Reduce the fraction and then simplify the remaining radical. The fraction does not reduce, so separate the radical and then simplify each remaining radical. 13 Example 4 – Solution Reduce the fraction. Since the fraction does not reduce further, separate the radical and simplify. Reduce the fraction. Simplify the radical. 14 Dividing Radicals and Rationalizing the Denominator Clearing any remaining radicals from the denominator of a fraction is called rationalizing the denominator. This process uses multiplication on the top and bottom of the fraction to force any radicals in the denominator to simplify completely. The key to rationalizing the denominator of a fraction is to multiply both the numerator and the denominator of the fraction by the right radical expression. 15 Example 5 – Rationalizing the denominator Rationalize the denominator and simplify the following radical expressions. 16 Example 5(a) – Solution Separate into two radicals. Multiply the numerator and denominator by the denominator. This is the same as multiplying by 1. Simplify the radicals. 17 Example 5(b) – Solution Multiply the numerator and denominator by the denominator. 18

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