Chapter 8 Section 2 Copyright 2008 Pearson Education

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

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

8. 2 1 2 3 4 5 Multiplying, Dividing, and Simplifying Radicals Multiply square

8. 2 1 2 3 4 5 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

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

Objective 1 Multiply square root radicals. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8. 2 - 3

Multiply square root radicals. For nonnegative real numbers a and b, and That is,

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. 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 Slide 8. 2 - 4

EXAMPLE 1 Using the Product Rule to Multiply Radicals Solution: Find each product. Assume

EXAMPLE 1 Using the Product Rule to Multiply Radicals Solution: Find each product. Assume that Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8. 2 - 5

Objective 2 Simplify radicals by using the product rule. Copyright © 2008 Pearson Education,

Objective 2 Simplify radicals by using the product rule. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8. 2 - 6

Simplify radicals using the product rule. A square root radical is simplified when no

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: Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8. 2 - 7

EXAMPLE 2 Using the Product Rule to Simplify Radicals Solution: Simplify each radical. Copyright

EXAMPLE 2 Using the Product Rule to Simplify Radicals Solution: Simplify each radical. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8. 2 - 8

EXAMPLE 3 Multiplying and Simplifying Radicals Solution: Find each product and simplify. Copyright ©

EXAMPLE 3 Multiplying and Simplifying Radicals Solution: Find each product and simplify. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8. 2 - 9

Objective 3 Simplify radicals by using the quotient rule. Copyright © 2008 Pearson Education,

Objective 3 Simplify radicals by using the quotient rule. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8. 2 - 10

Simplify radicals by using the quotient rule. The quotient rule for radicals is similar

Simplify radicals by using the quotient rule. The quotient rule for radicals is similar to the product rule. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8. 2 - 11

EXAMPLE 4 Using the Quotient Rule to Simply Radicals Solution: Simplify each radical. Copyright

EXAMPLE 4 Using the Quotient Rule to Simply Radicals Solution: Simplify each radical. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8. 2 - 12

EXAMPLE 5 Using the Quotient Rule to Divide Radicals Simplify. Solution: Copyright © 2008

EXAMPLE 5 Using the Quotient Rule to Divide Radicals Simplify. Solution: Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8. 2 - 13

EXAMPLE 6 Using Both the Product and Quotient Rules Simplify. Solution: Copyright © 2008

EXAMPLE 6 Using Both the Product and Quotient Rules Simplify. Solution: Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8. 2 - 14

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

Objective 4 Simplify radicals involving variables. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8. 2 - 15

Simplify radicals involving variables. Radicals can also involve variables. The square root of a

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 Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8. 2 - 16

EXAMPLE 7 Simplifying Radicals Involving Variables Simplify each radical. Assume that all variables represent

EXAMPLE 7 Simplifying Radicals Involving Variables Simplify each radical. Assume that all variables represent positive real numbers. Solution: Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8. 2 - 17

Objective 5 Simplify other roots. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson

Objective 5 Simplify other roots. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8. 2 - 18

Simplify other roots. To simplify cube roots, look for factors that are perfect cubes.

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, Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8. 2 - 19

EXAMPLE 8 Simplifying Other Roots Simplify each radical. Solution: Copyright © 2008 Pearson Education,

EXAMPLE 8 Simplifying Other Roots Simplify each radical. Solution: Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8. 2 - 20

Simplify other roots. (cont’d) Other roots of radicals involving variables can also be simplified.

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. Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8. 2 - 21

EXAMPLE 9 Simplifying Cube Roots Involving Variables Simplify each radical. Solution: Copyright © 2008

EXAMPLE 9 Simplifying Cube Roots Involving Variables Simplify each radical. Solution: Copyright © 2008 Pearson Education, Inc. Publishing as Pearson Addison-Wesley Slide 8. 2 - 22