Precalculus: Mathematics for Calculus, 7th Edition

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

Chapter 1 - Section 1.2 - Exponents and Radicals - 1.2 Exercises - Page 23: 81

Answer

$a)$ $\dfrac{\sqrt{5x}}{5x}$ $b)$ $\dfrac{\sqrt{5x}}{5}$ $c)$ $\dfrac{\sqrt[5]{x^{2}}}{x}$

Work Step by Step

$a)$ $\dfrac{1}{\sqrt{5x}}$ Multiply the numerator and the denominator by $\sqrt{5x}$ and simplify: $\dfrac{1}{\sqrt{5x}}=\dfrac{1}{\sqrt{5x}}\cdot\dfrac{\sqrt{5x}}{\sqrt{5x}}=\dfrac{\sqrt{5x}}{\sqrt{(5x)^{2}}}=\dfrac{\sqrt{5x}}{5x}$ $b)$ $\sqrt{\dfrac{x}{5}}$ Rewrite as $\dfrac{\sqrt{x}}{\sqrt{5}}$: $\sqrt{\dfrac{x}{5}}=\dfrac{\sqrt{x}}{\sqrt{5}}=...$ Multiply the numerator and the denominator by $\sqrt{5}$ and simplify: $...=\dfrac{\sqrt{x}}{\sqrt{5}}\cdot\dfrac{\sqrt{5}}{\sqrt{5}}=\dfrac{\sqrt{5x}}{\sqrt{5^{2}}}=\dfrac{\sqrt{5x}}{5}$ $c)$ $\sqrt[5]{\dfrac{1}{x^{3}}}$ Rewrite as $\dfrac{\sqrt[5]{1}}{\sqrt[5]{x^{3}}}$ and simplify: $\sqrt[5]{\dfrac{1}{x^{3}}}=\dfrac{\sqrt[5]{1}}{\sqrt[5]{x^{3}}}=\dfrac{1}{\sqrt[5]{x^{3}}}=...$ Multiply the numerator and the denominator by $\sqrt[5]{x^{2}}$ and simplify again: $...=\dfrac{1}{\sqrt[5]{x^{3}}}\cdot\dfrac{\sqrt[5]{x^{2}}}{\sqrt[5]{x^{2}}}=\dfrac{\sqrt[5]{x^{2}}}{\sqrt[5]{x^{5}}}=\dfrac{\sqrt[5]{x^{2}}}{x}$
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