Copyright 2006 Pearson Education Inc Publishing as Pearson
Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7 - 1
7 7. 1 7. 2 7. 3 7. 4 7. 5 7. 6 7. 7 7. 8 Exponents and Radicals Radical Expressions and Functions Rational Numbers as Exponents Multiplying Radical Expressions Dividing Radical Expressions Containing Several Radical Terms Solving Radical Equations Geometric Applications The Complex Numbers Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
7. 3 Multiplying Radical Expressions n Simplifying by Factoring n Multiplying and Simplifying Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley
Multiplying Radical Expressions Note that This example suggests the following. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7 - 4
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: Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7 - 5
Example Multiply. Solution Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7 - 6
Caution The product rule for radicals applies only when radicals have the same index: Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7 - 7
Simplifying by Factoring The number p is a perfect square if there exists a rational number q for which q 2 = p. We say that p is a perfect nth power if qn = p for some rational number q. The product rule allows us to simplify whenever ab contains a factor that is a perfect nth power. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7 - 8
Using The Product Rule to Simplify ( must both be real numbers. ) Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7 - 9
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. Take the nth root of each factor. 3. Simplification is complete when no radicand has a factor that is a perfect nth power. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7 - 10
It is often safe to assume that a radicand does not represent a negative number raised to an even power. We will make this assumption. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7 - 11
Example Simplify by factoring: Solution 100 is the largest perfectsquare factor of 300. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7 - 12
Solution continued 27 s 3 is the largest perfect third-power factor. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7 - 13
Remember: To simplify an nth root, identify factors in the radicand with exponents that are multiples of n. Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7 - 14
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 © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7 - 15
Example Multiply and simplify. Solution Multiplying radicands 4 is a perfect square Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7 - 16
Solution continued Copyright © 2006 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 7 - 17
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