90 CHAPTER R Review of Basic Concepts EXAMPLE 11 Simplifying Radical Expressions with Fractions Simplify each expression. Assume all variables represent positive real numbers. (a) 24 xy3 24 x3y2 (b) B3 5 x6 - B3 4 x9 SOLUTION (a) 24 xy3 24 x3y2 = B4 xy3 x3y2 Quotient rule = B4 y x2 Simplify the radicand. = 24 y 24 x2 Quotient rule = 24 y 24 x2 # 24 x2 24 x2 Rationalize the denominator. = 24 x2y x 24 x2 # 24 x2 = 24 x4 = x (b) B3 5 x6 - B3 4 x9 = 23 5 23 x6 - 23 4 23 x9 Quotient rule = 23 5 x2 - 23 4 x3 Simplify the denominators. = x23 5 x3 - 23 4 x3 Write with a common denominator. = x23 5 - 23 4 x3 Subtract the numerators. S Now Try Exercises 139 and 143. In Example 9(b), we saw that the product A 2 7 - 210 B A 27 + 210 B equals -3, a rational number. This suggests a way to rationalize a denominator that is a binomial in which one or both terms is a square root radical. The expressions a - b and a + b are conjugates. LOOKING AHEAD TO CALCULUS Another standard problem in calculus is investigating the value that an expression such as 2 x2 + 9 - 3 x2 approaches as x approaches 0. This cannot be done by simply substituting 0 for x because the result is 0 0 . However, by rationalizing the numerator, we can show that for x ≠0 the expression is equivalent to 1 2 x2 + 9 + 3 . Then, by substituting 0 for x, we find that the original expression approaches 1 6 as x approaches 0. 1 + 22 1 + 22 = 1 EXAMPLE 12 Rationalizing a Binomial Denominator Rationalize the denominator of 1 1 - 22 . SOLUTION 1 1 - 22 = 1 A1 + 22 B A1 - 22 B A1 + 22 B Multiply numerator and denominator by the conjugate of the denominator, 1 + 22. = 1 + 22 1 - 2 1x - y21x + y2 = x2 - y2 = 1 + 22 -1 Subtract. = -1 - 22 Divide by -1. S Now Try Exercise 149.
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