Chapter 7 - Section 7.5 - Multiplying and Dividing Radical Expressions - 7.5 Exercises - Page 476: 84

$\dfrac{20-4\sqrt{6}}{19}$

Work Step by Step

$\bf{\text{Solution Outline:}}$ To rationalize the given radical expression, $\dfrac{4}{5+\sqrt{6}} ,$ multiply the numerator and the denominator by the conjugate of the denominator. Then use special products to multiply the result. $\bf{\text{Solution Details:}}$ Multiplying the numerator and the denominator of the given expression by the conjugate of the denominator, the expression above is equivalent to \begin{array}{l}\require{cancel} \dfrac{4}{5+\sqrt{6}}\cdot\dfrac{5-\sqrt{6}}{5-\sqrt{6}} \\\\= \dfrac{4(5-\sqrt{6})}{(5+\sqrt{6})(5-\sqrt{6})} .\end{array} 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} \dfrac{4(5-\sqrt{6})}{(5)^2-(\sqrt{6})^2} \\\\= \dfrac{4(5-\sqrt{6})}{25-6} \\\\= \dfrac{4(5-\sqrt{6})}{19} .\end{array} Using the Distributive Property which is given by $a(b+c)=ab+ac,$ the expression above is equivalent to \begin{array}{l}\require{cancel} \dfrac{4(5)-4(\sqrt{6})}{19} \\\\= \dfrac{20-4\sqrt{6}}{19} .\end{array}

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