Intermediate Algebra for College Students (7th Edition)

Published by Pearson
ISBN 10: 0-13417-894-7
ISBN 13: 978-0-13417-894-3

Chapter 7 - Section 7.5 - Multiplying with More Than One Term and Rationalizing Denominators - Exercise Set - Page 550: 91


$\displaystyle \frac{4x+y+4\sqrt{xy}}{y-4x}$

Work Step by Step

We lose the square roots in the denominator by applying the difference of squares formula: $(a\sqrt{x}+b\sqrt{y})(a\sqrt{x}-b\sqrt{y})=(a\sqrt{x})^{2}-(b\sqrt{y})^{2}=a^{2}x-b^{2}y$ $\displaystyle \frac{2\sqrt{x}+\sqrt{y}}{\sqrt{y}-2\sqrt{x}}\color{red}{ \cdot\frac{\sqrt{y}+2\sqrt{x}}{\sqrt{y}+2\sqrt{x}} }\qquad$ (rationalize) $=\displaystyle \frac{(2\sqrt{x}+\sqrt{y})(\sqrt{y}+2\sqrt{x})}{(\sqrt{y})^{2}-(2\sqrt{x})^{2}}$ ... use FOIL for the numerator $=\displaystyle \frac{2\sqrt{xy}+4(\sqrt{x})^{2}+(\sqrt{y})^{2}+2\sqrt{xy}}{y-2^{2}x}$ $=\displaystyle \frac{4x+y+4\sqrt{xy}}{y-4x}$
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