Simplifying a Fractional Expression Containing Radicals Simplify 3 9 x 2 x 2 3

# Simplifying a fractional expression containing

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Simplifying a Fractional Expression Containing Radicals Simplify: 3 9 - x 2 + x 2 3 9 - x 2 9 - x 2 . EXAMPLE 11 1 x + 7 - 1 x 7 . h. = - 1 x 1 x + h 2 , h Z 0, x Z 0, x Z - h x - x - h = - h. = - h 1 h 1 x 1 x + h 2 = x - x - h hx 1 x + h 2 1 x # x 1 x + h 2 = x + h. 1 x + h x 1 x + h 2 = x = x - 1 x + h 2 hx 1 x + h 2 = 1 x + h # x 1 x + h 2 - 1 x # x 1 x + h 2 h x 1 x + h 2 x 1 x + h 2 , h Z 0, x Z 0, x Z - h. = a 1 x + h - 1 x b x 1 x + h 2 h x 1 x + h 2 1 x + h - 1 x h x 1 x + h 2 . h 1 x + h , 1 x , 1 x + h - 1 x h . EXAMPLE 10 Simplify fractional expressions that occur in calculus.
78 Chapter P Prerequisites: Fundamental Concepts of Algebra Solution The least common denominator is Multiply the numerator and the denominator by Use the distributive property in the numerator. In the denominator: Because the original expression was in radical form, write the denominator in radical form. Check Point 11 Simplify: Another fractional expression that you will encounter in calculus is Can you see that this expression is not defined if However, in calculus, you will ask the following question: What happens to the expression as takes on values that get closer and closer to 0, such as and so on? The question is answered by first rationalizing the numerator .This process involves rewriting the fractional expression as an equivalent expression in which the numer- ator no longer contains any radicals. To rationalize a numerator, multiply by 1 to eliminate the radicals in the numerator . Multiply the numerator and the denominator by the conjugate of the numerator. Rationalizing a Numerator Rationalize the numerator: Solution The conjugate of the numerator is If we multiply the numerator and denominator by the simplified numerator will not contain a radical.Therefore, we multiply by 1, choosing for 1. 1 x + h + 1 x 1 x + h + 1 x 1 x + h + 1 x , 1 x + h + 1 x . 2 x + h - 1 x h . EXAMPLE 12 h = 0.1, h = 0.01, h = 0.001, h = 0.0001, h h = 0? 2 x + h - 1 x h . 1 x + 1 1 x x . = 9 3 1 9 - x 2 2 3 = 1 9 - x 2 2 3 2 . 1 9 - x 2 2 1 1 9 - x 2 2 1 2 = 1 9 - x 2 2 1 + 1 2 = 1 9 - x 2 2 + x 2 1 9 - x 2 2 3 2 = 3 9 - x 2 3 9 - x 2 + x 2 3 9 - x 2 3 9 - x 2 1 9 - x 2 2 3 9 - x 2 3 9 - x 2 . = 3 9 - x 2 + x 2 3 9 - x 2 9 - x 2 # 3 9 - x 2 3 9 - x 2 3 9 - x 2 . 3 9 - x 2 + x 2 3 9 - x 2 9 - x 2 Rationalize numerators.
Section P.6 Rational Expressions 79 Multiply by 1. and Simplify: Divide both the numerator and denominator by What happens to as gets closer and closer to 0? In Example 12, we showed that As gets closer to 0, the expression on the right gets closer to or Thus, the fractional expression approaches as gets closer to 0. Check Point 12 Rationalize the numerator: 2 x + 3 - 1 x 3 . h 1 2 1 x 2 x + h - 1 x h 1 2 1 x . 1 1 x + 1 x , 1 2 x + 0 + 1 x = h 2 x + h - 1 x h = 1 2 x + h + 1 x . h 2 x + h - 1 x h h . = 1 2 x + h + 1 x , h Z 0 x + h - x = h . = h h A 2 x + h + 1 x B 1 1 x 2 2 = x . A 2 x + h B 2 = x + h = x + h - x h A 2 x + h + 1 x B A 1 a B 2 - A 2 b B 2 A 1 a - 2 b B A 1 a + 2 b B = = A 2 x + h B 2 - 1 1 x 2 2 h A 2 x + h + 1 x B 2 x + h - 1 x h = 2 x + h - 1 x h # 2 x + h + 1 x 2 x + h + 1 x Exercise Set P.6 Practice Exercises In Exercises 1–6, find all numbers that must be excluded from the domain of each rational expression.