Answer
$\dfrac{x-2\sqrt{xy}+y}{x-y}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To rationalize the given radical expression, $
\dfrac{\sqrt{x}-\sqrt{y}}{\sqrt{x}+\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{\sqrt{x}-\sqrt{y}}{\sqrt{x}+\sqrt{y}}\cdot\dfrac{\sqrt{x}-\sqrt{y}}{\sqrt{x}-\sqrt{y}}
\\\\=
\dfrac{(\sqrt{x}-\sqrt{y})(\sqrt{x}-\sqrt{y})}{(\sqrt{x}+\sqrt{y})(\sqrt{x}-\sqrt{y})}
\\\\=
\dfrac{(\sqrt{x}-\sqrt{y})^2}{(\sqrt{x}+\sqrt{y})(\sqrt{x}-\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{(\sqrt{x}-\sqrt{y})^2}{(\sqrt{x})^2-(\sqrt{y})^2}
\\\\=
\dfrac{(\sqrt{x}-\sqrt{y})^2}{x-y}
.\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}
\dfrac{(\sqrt{x})^2-2(\sqrt{x})(\sqrt{y})+(\sqrt{y})^2}{x-y}
\\\\=
\dfrac{x-2(\sqrt{x})(\sqrt{y})+y}{x-y}
.\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}
\dfrac{x-2\sqrt{x(y)}+y}{x-y}
\\\\=
\dfrac{x-2\sqrt{xy}+y}{x-y}
.\end{array}
