Copyright © Cengage Learning. All rights reserved. 8 Radical Functions

Copyright © Cengage Learning. All rights reserved. 8.3 Multiplying and Dividing Radicals

3 Objectives  Multiply radical expressions.  Divide radical expressions.  Use conjugates to rationalize denominators.

4 Multiplying Radicals

5 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.

6 Example 1 – Multiplying radicals Multiply the following and simplify the result.

7 Example 1 – Solution There are no square factors, so it is simplified. 20 has a perfect square factor, so simplify.

8 Example 1 – Solution 36 and a 2 are perfect squares, so simplify. 20 has no perfect cube factors, but the m 4 can be simplified. cont’d

9 Multiplying Radicals If the expressions are more complicated, you will multiply the coefficients of the radicals together and multiply the radicands together. Once you have multiplied each of these together, you should simplify the result.

10 Dividing Radicals and Rationalizing the Denominator

11 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.

12 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.

13 Example 4 – Simplifying radicals with division Simplify the following radicals.

14 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.

15 Example 4 – Solution Reduce the fraction. Since the fraction does not reduce further, separate the radical and simplify. Reduce the fraction. Simplify the radical.

16 Dividing Radicals and Rationalizing the Denominator In the previous example all of the denominators simplified to the point at which no radicals remain. This will not always happen, but mathematicians often like to have no radicals in the denominator of a fraction. Having no radicals in a denominator makes some operations easier and is considered a standard way to write a simplified fraction.

17 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.

18 Dividing Radicals and Rationalizing the Denominator This will allow the resulting denominator to simplify and be without any remaining radicals. With a single square root, rationalizing the denominator is usually accomplished by multiplying the numerator and denominator by the radical factor of the denominator.

19 Example 5 – Rationalizing the denominator Rationalize the denominator and simplify the following radical expressions.

20 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.

21 Example 5(b) – Solution Multiply the numerator and denominator by the denominator.

22 Example 5(c) – Solution Reduce the radicands. Multiply the numerator and denominator by the radical factor of the denominator. Simplify the radicals. Reduce the fraction.

23 Dividing Radicals and Rationalizing the Denominator

24 Conjugates

25 Conjugates

26 Conjugates Often fractions will have square roots in the denominator with another term. This requires you to use a different approach to rationalizing the denominator. If there are two terms, you will multiply by the conjugate of the denominator to clear all of the radicals in the denominator.

27 Example 7 – Using conjugates to rationalize denominators Rationalize the denominator of the following fractions.

28 Example 7(a) – Solution Multiply the numerator and denominator by the conjugate of the denominator.

29 Example 7(b) – Solution Multiply the numerator and denominator by the conjugate of the denominator. Use the distributive property. Simplify the radicals.

30 Example 7(c) – Solution Multiply the numerator and denominator by the conjugate of the denominator. Use the distributive property. Simplify the radicals and combine like terms.