Algebra: A Combined Approach (4th Edition)

Published by Pearson
ISBN 10: 0321726391
ISBN 13: 978-0-32172-639-1

Chapter 10 - Review: 127

Answer

$2i\sqrt{2}$

Work Step by Step

$\bf{\text{Solution Outline:}}$ To simplify the given expression, $ \sqrt{-8} ,$ use the properties of radicals and the definition of an imaginary number. $\bf{\text{Solution Details:}}$ Using the Product Rule of radicals which is given by $\sqrt[m]{x}\cdot\sqrt[m]{y}=\sqrt[m]{xy},$ the expression above is equivalent to\begin{array}{l}\require{cancel} \sqrt{-1}\cdot\sqrt{8} .\end{array} Using the definition of an imaginary number which is given by $i=\sqrt{-1},$ the expression above is equivalent to \begin{array}{l}\require{cancel} i\cdot\sqrt{8} .\end{array} Simplifying the radical by extracting the root of the factor that is a perfect power of the index results to \begin{array}{l}\require{cancel} i\cdot\sqrt{4\cdot2} \\\\= i\cdot\sqrt{(2)^2\cdot2} \\\\= i\cdot2\sqrt{2} \\\\= 2i\sqrt{2} .\end{array}
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