We will now use this property to multiply and divide radical expressions.

1. Earlier in the chapter, we used the following property to write radical expressions in simplest radical form: We will now use this property to multiply and divide radical expressions. Multiplying Radicals We can multiply radicals if their indexes are the same. If the radicals are both square roots, then they can be multiplied. Or if they are both cube roots, they can be multiplied. If the indexes are different, the radicals may not be multiplied. Of course, we always need to leave the answer in simplest radical form. Answer Solution: Your Turn Problem #1 1. Multiply radicands (both radicals have an index=2) 2. Write in simplest radical form.

2. Directions 1. Multiply whole numbers. Write the result in front of the radical. Multiply radicands. Write the result inside the radical. Solution: 2. Write in simplest radical form (if possible). Solution: Your Turn Problem #2 Multiply the following. Answers

3. Directions: 1. Multiply whole numbers. Write the result in front of the radical. Multiply radicands. Write the result inside the radical. Solution: 2. Write in simplest radical form (if possible). Answers Solution: Your Turn Problem #3 Multiply the following.

4. Solution: Directions 1. Multiply the radicands since the indexes are the same. 2. Write in simplest radical form. Solution: Your Turn Problem #4 Multiply the following. Answers

5. Our next objective is to multiply a radical expression by a sum or difference of radical expressions. This will require the use of the distributive property. Solution: 1. Distribute. 2. Write each expression in simplest radical form. Your Turn Problem #5 Answer:

6. Solution: 1. Distribute. 2. Write each expression in simplest radical form. Solution: Answers Your Turn Problem #6 Multiply the following.

7. 1. Distribute. Solution: 1. Distribute. 2. Write each expression in simplest radical form. Solution: 2. Write each expression in simplest radical form. Your Turn Problem #7 Multiply. Answer: Your Turn Problem #8 Answer Multiply.

8. Our next objective is to multiply binomials which contain radicals. Remember to to use the foil process when multiplying binomials. Solution: Answer Your Turn Problem #9 Multiply. Solution: Your Turn Problem #10 Multiply. This is an example of multiplying two conjugates, a + b and a – b. The product of two conjugates will always equal an integer. Answer: 47

9. Rationalizing Binomial Denominators Containing Square Roots In a previous section, we learned how to “get rid” of the radical in the denominator (i.e. rationalize the denominator). If the radical is a square root, the procedure was to multiply the denominator by itself. Of course, whatever you multiply to the denominator, you need to multiply to the numerator. Answer Your Turn Problem #11 We are now going to rationalize the denominator when the denominator is a binomial. We will use the fact: multiplying conjugates will result in an integer. Directions: Multiply the both numerator and denominator by the conjugate of the denominator. Then simplify.

10. Before we do a few more examples, lets practice simplifying rational expressions. 4 3 Informal Method Find the greatest common factor for all of the terms. (What # divides evenly into all terms?) 1 2 Answer = 3. Then divide all terms by 3. 2 Same result. Find the greatest common factor for all of the terms. (What # divides evenly into all terms?) 4 5 Answer = No GCF. Cannot be simplified further. 2 First we can factor out the greatest common factor in the numerator. Next, reduce. Find the greatest common factor for all of the terms. (What # divides evenly into all terms?) Answer = 6. Then divide all terms by 6.

11. Answer Your Turn Problem #12 Directions: Multiply both numerator and denominator by the conjugate of the denominator. Then simplify.

12. Answer: Your Turn Problem #13

13. For our last example, we’ll do one that contains variables. Answer: Your Turn Problem #14 The End B.R. 10-19-06