Answer
$\dfrac{4\sqrt{x}+8\sqrt{y}}{x-4y}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To rationalize the given radical expression, $
\dfrac{4}{\sqrt{x}-2\sqrt{y}}
,$ 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 of the given expression by the conjugate of the denominator, the expression above is equivalent to
\begin{array}{l}\require{cancel}
\dfrac{4}{\sqrt{x}-2\sqrt{y}} \cdot\dfrac{\sqrt{x}+2\sqrt{y}}{\sqrt{x}+2\sqrt{y}}
\\\\=
\dfrac{4(\sqrt{x}+2\sqrt{y})}{(\sqrt{x}-2\sqrt{y})(\sqrt{x}+2\sqrt{y})}
.\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(\sqrt{x}+2\sqrt{y})}{(\sqrt{x})^2-(2\sqrt{y})^2}
\\\\=
\dfrac{4(\sqrt{x}+2\sqrt{y})}{x-4y}
.\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(\sqrt{x})+4(2\sqrt{y})}{x-4y}
\\\\=
\dfrac{4\sqrt{x}+8\sqrt{y}}{x-4y}
.\end{array}
