Precalculus: Mathematics for Calculus, 7th Edition

Published by Brooks Cole
ISBN 10: 1305071751
ISBN 13: 978-1-30507-175-9

Chapter 1 - Section 1.4 - Rational Expressions - 1.4 Exercises: 68

Answer

$x-\dfrac{y}{\dfrac{x}{y}+\dfrac{y}{x}}=\dfrac{x^{3}}{x^{2}+y^{2}}$

Work Step by Step

$x-\dfrac{y}{\dfrac{x}{y}+\dfrac{y}{x}}$ Evaluate the sum in the denominator: $x-\dfrac{y}{\dfrac{x}{y}+\dfrac{y}{x}}=x-\dfrac{y}{\Big(\dfrac{x^{2}+y^{2}}{xy}\Big)}=...$ Evaluate the division: $...=x-\dfrac{(xy)y}{x^{2}+y^{2}}=x-\dfrac{xy^{2}}{x^{2}+y^{2}}=...$ Finally, evaluate the substraction and simplify: $...=\dfrac{x(x^{2}+y^{2})-xy^{2}}{x^{2}+y^{2}}=\dfrac{x^{3}+xy^{2}-xy^{2}}{x^{2}+y^{2}}=\dfrac{x^{3}}{x^{2}+y^{2}}$
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