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Copyright © 2012 Pearson Education, Inc. Slide 10- 1 7.3 Multiplying Radical Expressions ■ ■ ■ Multiplying Radical Expressions Simplifying by Factoring Multiplying and Simplifying Copyright © 2012 Pearson Education, Inc. Multiplying Radical Expressions Note that 4 9 2 3 6. 4 9 36 6. This example suggests the following. Copyright © 2012 Pearson Education, Inc. Slide 7- 3 The Product Rule for Radicals For any real numbers n a and n b , n a n b n a b. (The product of two nth roots is the nth root of the product of the two radicands.) Rational exponents can be used to derive this rule: n a n b a1/ n b1/ n a b 1/ n n a b. Copyright © 2012 Pearson Education, Inc. Slide 7- 4 Example Multiply. a) 5 6 Solution a) 5 6 5 6 30. b) 3 7 3 9 b) 3 7 3 9 3 7 9 3 63 x 5 c) 4 4 3 z x 5 x 5 5 x c) 4 4 4 4 3 z 3 z 3z Copyright © 2012 Pearson Education, Inc. Slide 7- 5 CAUTION! The product rule for radicals applies only when radicals have the same index: n a m b nm a b . Copyright © 2012 Pearson Education, Inc. Slide 7- 6 Simplifying by Factoring An integer p is a perfect square if there exists a rational number q for which q2 = p. We say that p is a perfect cube if q3 = p for some rational number q. In general, p is the perfect nth power if qn = p for some rational number q. The product rule allows us to simplify when a or b is a perfect nth power. Copyright © 2012 Pearson Education, Inc. n ab Slide 7- 7 Using The Product Rule to Simplify n ab n a n b ( n a and n b must both be real numbers.) Copyright © 2012 Pearson Education, Inc. Slide 7- 8 To Simplify a Radical Expression with Index n by Factoring 1. Express the radicand as a product in which one factor is the largest perfect nth power possible. 2. Rewrite the expression as the nth root of each factor. 3. Simplify the expression containing the perfect nth power. 4. Simplification is complete when no radicand has a factor that is a perfect nth power. Copyright © 2012 Pearson Education, Inc. Slide 7- 9 It is often safe to assume that a radicand does not represent a negative number raised to an even power. We will henceforth make this assumption ––unless functions are involved –– and discontinue use of absolutevalue notation when taking even roots. Copyright © 2012 Pearson Education, Inc. Slide 7- 10 Example Simplify by factoring: a) 300 b) 8m 4 n c) 3 54 s 4 Solution a) 300 100 3 100 is the largest perfectsquare factor of 300. 100 3 10 3 Copyright © 2012 Pearson Education, Inc. Slide 7- 11 Solution continued 8m4n 4 2 m4 n b) 4m 4 2n 2m c) 3 2 2n 3 54s 4 27 2 s3 s 27s3 is the largest perfect third-power factor. 3 27 s3 3 2s 3s 3 2s Copyright © 2012 Pearson Education, Inc. Slide 7- 12 Example If f x 2x2 12x 18, find a simplified form for f(x). Solution f x 2 x2 12 x 18 2 x2 6 x 9 x 3 x 3 2 2 2 x3 2 2 Factoring into two radicals. Taking the square root. Copyright © 2012 Pearson Education, Inc. Slide 7- 13 continued We can check by graphing y1 2 x2 12 x 18 and y2 x 3 2. Copyright © 2012 Pearson Education, Inc. Slide 7- 14 To simplify an nth root, identify factors in the radicand with exponents that are multiples of n. Copyright © 2012 Pearson Education, Inc. Slide 7- 15 Multiplying and Simplifying We have used the product rule for radicals to find products and also to simplify radical expressions. For some radical expressions, it is possible to do both: First find a product and then simplify. Copyright © 2012 Pearson Education, Inc. Slide 7- 16 Example Multiply and simplify. a) 10 6 3 23 4 7 3 b) 9 x y 9 x y Solution a) 10 6 60 4 15 Multiplying radicands 4 is a perfect square. 2 15 Copyright © 2012 Pearson Education, Inc. Slide 7- 17 Solution continued 3 23 4 7 3 7 9 3 b) 9 x y 9 x y 81x y 3 27 3 x6 x y9 3 6 3 9 3 3 27 x y 3x 3x 2 y3 3 3x Copyright © 2012 Pearson Education, Inc. Slide 7- 18