1. Slide 10- 1 Copyright © 2012 Pearson Education, Inc.

2. 7.5 Expressions Containing Several Radical Terms ■ Adding and Subtracting Radical Expressions ■ Products and Quotients of Two or More Radical Terms ■ Rationalizing Denominators and Numerators with Two Terms ■ Terms with Differing Indices

3. Slide 7- 3 Copyright © 2012 Pearson Education, Inc. Adding and Subtracting Radical Expressions When two radical expressions have the same indices and radicands, they are said to be like radicals. Like radicals can be combined (added or subtracted) in much the same way that we combined like terms earlier in this text.

4. Slide 7- 4 Copyright © 2012 Pearson Education, Inc. Example Solution Simplify by combining like radical terms.

5. Slide 7- 5 Copyright © 2012 Pearson Education, Inc. Example Solution Simplify by combining like radical terms.

6. Slide 7- 6 Copyright © 2012 Pearson Education, Inc. Products and Quotients of Two or More Radical Terms Radical expressions often contain factors that have more than one term. The procedure for multiplying out such expressions is similar to finding products of polynomials. Some products will yield like radical terms, which we can now combine.

7. Slide 7- 7 Copyright © 2012 Pearson Education, Inc. Example Solution Multiply. Using the distributive law

8. Slide 7- 8 Copyright © 2012 Pearson Education, Inc. Solution F O I L

9. Slide 7- 9 Copyright © 2012 Pearson Education, Inc. In part (c) of the last example, notice that the inner and outer products in FOIL are opposites, the result, m – n, is not itself a radical expression. Pairs of radical terms like, are called conjugates.

10. Slide 7- 10 Copyright © 2012 Pearson Education, Inc. Rationalizing Denominators and Numerators with Two Terms The use of conjugates allows us to rationalize denominators or numerators with two terms.

11. Slide 7- 11 Copyright © 2012 Pearson Education, Inc. Example Solution Rationalize the denominator: Multiplying by 1 using the conjugate

12. Slide 7- 12 Copyright © 2012 Pearson Education, Inc. Example Solution Rationalize the denominator:

13. Slide 7- 13 Copyright © 2012 Pearson Education, Inc. To rationalize a numerator with more than one term, we use the conjugate of the numerator.

14. Slide 7- 14 Copyright © 2012 Pearson Education, Inc. Example Solution Rationalize the numerator:

15. Slide 7- 15 Copyright © 2012 Pearson Education, Inc. Terms with Differing Indices To multiply or divide radical terms with different indices, we can convert to exponential notation, use the rules for exponents, and then convert back to radical notation.

16. Slide 7- 16 Copyright © 2012 Pearson Education, Inc. To Simplify Products or Quotients with Differing Indices 1. Convert all radical expressions to exponential notation. 2. When the bases are identical, subtract exponents to divide and add exponents to multiply. This may require finding a common denominator. 3. Convert back to radical notation and, if possible, simplify.

17. Slide 7- 17 Copyright © 2012 Pearson Education, Inc. Example Solution Multiply and, if possible, simplify: Converting to exponential notation Adding exponents Converting to radical notation Simplifying

18. Slide 7- 18 Copyright © 2012 Pearson Education, Inc. Example Solution Divide and, if possible, simplify: