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7.3 Multiplying Radical Expressions ■ Multiplying Radical Expressions ■ Simplifying by Factoring ■ Multiplying and Simplifying

Slide 7- 3 Copyright © 2012 Pearson Education, Inc. Multiplying Radical Expressions Note that This example suggests the following.

Slide 7- 4 Copyright © 2012 Pearson Education, Inc. The Product Rule for Radicals For any real numbers (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:

Slide 7- 5 Copyright © 2012 Pearson Education, Inc. Example Multiply. Solution

Slide 7- 6 Copyright © 2012 Pearson Education, Inc. CAUTION! The product rule for radicals applies only when radicals have the same index:

Slide 7- 7 Copyright © 2012 Pearson Education, Inc. Simplifying by Factoring An integer p is a perfect square if there exists a rational number q for which q 2 = p. We say that p is a perfect cube if q 3 = p for some rational number q. In general, p is the perfect nth power if q n = p for some rational number q. The product rule allows us to simplify when a or b is a perfect nth power.

Slide 7- 8 Copyright © 2012 Pearson Education, Inc. Using The Product Rule to Simplify ( must both be real numbers.)

Slide 7- 9 Copyright © 2012 Pearson Education, Inc. 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.

Slide 7- 10 Copyright © 2012 Pearson Education, Inc. 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 absolute- value notation when taking even roots.

Slide 7- 11 Copyright © 2012 Pearson Education, Inc. Example Solution Simplify by factoring: 100 is the largest perfect- square factor of 300.

Slide 7- 12 Copyright © 2012 Pearson Education, Inc. Solution continued 27s 3 is the largest perfect third-power factor.

Slide 7- 13 Copyright © 2012 Pearson Education, Inc. Example Solution Factoring into two radicals. If find a simplified form for f(x). Taking the square root.

Slide 7- 14 Copyright © 2012 Pearson Education, Inc. continued We can check by graphing

Slide 7- 15 Copyright © 2012 Pearson Education, Inc. To simplify an nth root, identify factors in the radicand with exponents that are multiples of n.

Slide 7- 16 Copyright © 2012 Pearson Education, Inc. 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. Multiplying and Simplifying

Slide 7- 17 Copyright © 2012 Pearson Education, Inc. Example Solution Multiply and simplify. Multiplying radicands 4 is a perfect square.