Chapter 6 6-2 Rational Exponents

Objectives Evaluate and simplify expressions containing rational exponents.

Index Recall that the radical symbol is used to indicate roots. The index is the small number to the left of the radical symbol that tells which root to take. For example represents a cubic root. Since 23 = 222 = 8,

Writing rational expressions Another way to write nth roots is by using fractional exponents. For example, for b >1, suppose Solution: Square both sides b1 = b2k If bm = bn, then m = n. 1 = 2k So for all b > 1,

Definition of rational expression

Example#1 simplifying A) Solution: B) = 7 = 2 + 3 = 5

Example#1A Simplify each expression. A) Solution: = 3 B) Solution: = 11 + 4 = 15

Rational exponents A fractional exponent can have a numerator other than 1, as in the expression . You can write the exponent as a product in two different ways. P Power of a Power Property Or Definiti Definition of 1/b

Definition

Example 2: Simplifying Expressions with Fractional Exponents Simplify each expression. A) Solution: = 3 5 = 243 B) Solution: = 5 2 =25

Example#2A Simplify each expression. A) Solution: B) = 8

Additional Example 3: Application Given a cube with surface area S, the volume V of the cube can be found by using the formula Find the volume of a cube with surface area 54 m2

Solution: Substitute 54 for s. Simplify inside the parentheses. The volume of the cube is 27 m3.

Sign of rational expressions Remember that always indicates a nonnegative square root. When you simplify variable expressions that contain , such as the answer cannot be negative. But x may be negative. Therefore you simplify as |x| to ensure the answer is nonnegative.

Additional Example 4: Properties of Exponents to Simplify Expressions Simplify. All variables represent nonnegative numbers. Solution: •

Example 4A Simplify. All variables represent nonnegative numbers. Solution:

Example 4B Simplify. All variables represent nonnegative numbers. Solution: = xy

Student guided practice Do even problems 2-25 in your book page 401

Homework Do even problems in your book 31-54 in your book page 401

Closure Today we learned about rational expressions Next class we are going to learn about polynomials