Answer
$4\sqrt{7}-4\sqrt{5}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To rationalize the given radical expression, $
\dfrac{8}{\sqrt{7}+\sqrt{5}}
,$ multiply the numerator and the denominator by the conjugate of the denominator. Then use special products to simplify the result.
$\bf{\text{Solution Details:}}$
Multiplying the numerator and the denominator by the conjugate of the denominator results to
\begin{array}{l}\require{cancel}
\dfrac{8}{\sqrt{7}+\sqrt{5}}\cdot\dfrac{\sqrt{7}-\sqrt{5}}{\sqrt{7}-\sqrt{5}}
\\\\=
\dfrac{8(\sqrt{7}-\sqrt{5})}{(\sqrt{7}+\sqrt{5})(\sqrt{7}-\sqrt{5})}
.\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{8(\sqrt{7}-\sqrt{5})}{(\sqrt{7})^2-(\sqrt{5})^2}
\\\\=
\dfrac{8(\sqrt{7}-\sqrt{5})}{7-5}
\\\\=
\dfrac{8(\sqrt{7}-\sqrt{5})}{2}
\\\\=
\dfrac{\cancel{2}(4)(\sqrt{7}-\sqrt{5})}{\cancel{2}}
\\\\=
4(\sqrt{7}-\sqrt{5})
.\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}
4(\sqrt{7})+4(-\sqrt{5})
\\\\=
4\sqrt{7}-4\sqrt{5}
.\end{array}
