Chapter 8 Section 2 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley

Multiplying, Dividing, and Simplifying Radicals Multiply square root radicals. Simplify radicals by using the product rule. Simplify radicals by using the quotient rule. Simplify radicals involving variables. Simplify other roots

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 1 Objective 1 Multiply square root radicals. Slide

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Multiply square root radicals. For nonnegative real numbers a and b, and That is, the product of two square roots is the square root of the product, and the square root of a product is the product of the square roots. Slide It is important to note that the radicands not be negative numbers in the product rule. Also, in general,

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 1 Find each product. Assume that Using the Product Rule to Multiply Radicals Slide Solution:

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 2 Objective 2 Simplify radicals by using the product rule. Slide

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Simplify radicals using the product rule. A square root radical is simplified when no perfect square factor remains under the radical sign. This can be accomplished by using the product rule: Slide

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 2 Simplify each radical. Using the Product Rule to Simplify Radicals Slide Solution:

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 3 Find each product and simplify. Multiplying and Simplifying Radicals Slide Solution:

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 3 Objective 3 Slide Simplify radicals by using the quotient rule.

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Simplify radicals by using the quotient rule. The quotient rule for radicals is similar to the product rule. Slide

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 4 Simplify each radical. Solution: Using the Quotient Rule to Simply Radicals Slide

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 5 Simplify. Solution: Using the Quotient Rule to Divide Radicals Slide

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Simplify. EXAMPLE 6 Using Both the Product and Quotient Rules Slide Solution:

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 4 Objective 4 Slide Simplify radicals involving variables.

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Simplify radicals involving variables. Radicals can also involve variables. The square root of a squared number is always nonnegative. The absolute value is used to express this. The product and quotient rules apply when variables appear under the radical sign, as long as the variables represent only nonnegative real numbers Slide

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 7 Simplifying Radicals Involving Variables Slide Simplify each radical. Assume that all variables represent positive real numbers. Solution:

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley 5 Objective 5 Simplify other roots. Slide

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Simplify other roots. To simplify cube roots, look for factors that are perfect cubes. A perfect cube is a number with a rational cube root. For example,, and because 4 is a rational number, 64 is a perfect cube. For all real number for which the indicated roots exist, Slide

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 8 Simplifying Other Roots Slide Simplify each radical. Solution:

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Simplify other roots. (cont’d) Other roots of radicals involving variables can also be simplified. To simplify cube roots with variables, use the fact that for any real number a, This is true whether a is positive or negative. Slide

Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley EXAMPLE 9 Simplifying Cube Roots Involving Variables Slide Simplify each radical. Solution: