Intermediate Algebra (12th Edition)

Published by Pearson
ISBN 10: 0321969359
ISBN 13: 978-0-32196-935-4

Chapter 7 - Section 7.5 - Multiplying and Dividing Radical Expressions - 7.5 Exercises: 41

Answer

$-1+2\sqrt{6}$

Work Step by Step

$\bf{\text{Solution Outline:}}$ To simplify the given radical expression, $ [(\sqrt{2}+\sqrt{3})-\sqrt{6}][(\sqrt{2}+\sqrt{3})+\sqrt{6}] ,$ use the special product on multiplying the sum and difference of like terms. $\bf{\text{Solution Details:}}$ Using the product of the sum and difference of like terms which is given by $(a+b)(a-b)=a^2-b^2,$ the expression above is equivalent \begin{array}{l}\require{cancel} (\sqrt{2}+\sqrt{3})^2-(\sqrt{6})^2 \\\\= (\sqrt{2}+\sqrt{3})^2-6 .\end{array} Using the square of a binomial which is given by $(a+b)^2=a^2+2ab+b^2$ or by $(a-b)^2=a^2-2ab+b^2,$ the expression above is equivalent to \begin{array}{l}\require{cancel} [(\sqrt{2})^2+2(\sqrt{2})(\sqrt{3})+(\sqrt{3})^2]-6 \\\\= 2+2(\sqrt{2})(\sqrt{3})+3-6 .\end{array} 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} 2+2\sqrt{2(3)}+3-6 \\\\= 2+2\sqrt{6}+3-6 .\end{array} Combining like terms, the expression above is equivalent to \begin{array}{l}\require{cancel} (2+3-6)+2\sqrt{6} \\\\= -1+2\sqrt{6} .\end{array}
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