College Algebra (11th Edition)

Published by Pearson
ISBN 10: 0321671791
ISBN 13: 978-0-32167-179-0

Chapter R - Section R.7 - Radical Expressions - R.7 Exercises - Page 68: 87

Answer

$\dfrac{x\sqrt[3]{2}-\sqrt[3]{5}}{x^3}$

Work Step by Step

$\bf{\text{Solution Outline:}}$ Simplify each term in the given expression, $ \sqrt[3]{\dfrac{2}{x^6}}-\sqrt[3]{\dfrac{5}{x^9}} .$ Then make the terms similar (same denominator) to combine the numerators. $\bf{\text{Solution Details:}}$ Simplifying each term of the expression above results to \begin{array}{l}\require{cancel} \sqrt[3]{\dfrac{1}{x^6}\cdot2}-\sqrt[3]{\dfrac{1}{x^9}\cdot5} \\\\= \sqrt[3]{\left(\dfrac{1}{x^2}\right)^3\cdot2}-\sqrt[3]{\left(\dfrac{1}{x^3}\right)^3\cdot5} \\\\= \dfrac{1}{x^2}\sqrt[3]{2}-\dfrac{1}{x^3}\sqrt[3]{5} \\\\= \dfrac{\sqrt[3]{2}}{x^2}-\dfrac{\sqrt[3]{5}}{x^3} .\end{array} To simplify the expression above, make the terms similar by multiplying the necessary term/s to an expression equal to $1$. Hence, the expression above is equivalent to \begin{array}{l}\require{cancel} \dfrac{\sqrt[3]{2}}{x^2}\cdot\dfrac{x}{x}-\dfrac{\sqrt[3]{5}}{x^3} \\\\= \dfrac{x\sqrt[3]{2}}{x^3}-\dfrac{\sqrt[3]{5}}{x^3} .\end{array} To combine similar terms, add/subtract the numerators and copy the similar denominator. Hence, the expression above is equivalent to \begin{array}{l}\require{cancel} \dfrac{x\sqrt[3]{2}-\sqrt[3]{5}}{x^3} .\end{array}
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