Answer
$\dfrac{\sqrt{3}-\sqrt{7}}{2}$
Work Step by Step
Multiplying the numerator and the denominator by the conjugate of the denominator, the given expression is equivalent to:
\begin{align*}\require{cancel}
&
=\dfrac{5-\sqrt{21}}{\sqrt{3}-\sqrt{7}}\cdot\dfrac{\sqrt{3}+\sqrt{7}}{\sqrt{3}+\sqrt{7}}
\\\\&=
\dfrac{(5-\sqrt{21})(\sqrt{3}+\sqrt{7})}{(\sqrt{3})^2-(\sqrt{7})^2}
&*\left( \text{use }(a+b)(a-b)=a^2-b^2 \right)
\\\\&=
\dfrac{5(\sqrt{3})+5(\sqrt{7})-\sqrt{21}(\sqrt{3})-\sqrt{21}(\sqrt{7})}{(\sqrt{3})^2-(\sqrt{7})^2}
&\left( \text{use FOIL} \right)
\\\\&=
\dfrac{5\sqrt{3}+5\sqrt{7}-\sqrt{63}-\sqrt{147}}{3-7}
\\\\&=
\dfrac{5\sqrt{3}+5\sqrt{7}-\sqrt{9\cdot7}-\sqrt{49\cdot3}}{-4}
&\left( \text{extract perfect powers of index} \right)
\\\\&=
\dfrac{5\sqrt{3}+5\sqrt{7}-3\sqrt{7}-7\sqrt{3}}{-4}
\\\\&=
\dfrac{(5\sqrt{3}-7\sqrt{3})+(5\sqrt{7}-3\sqrt{7})}{-4}
&\left( \text{combine like terms} \right)
\\\\&=
\dfrac{-2\sqrt{3}+2\sqrt{7}}{-4}
\\\\&=
\dfrac{\cancel{-2}^1\sqrt{3}\cancel{+2}^{-1}\sqrt{7}}{\cancel{-4}^2}
&\left( \text{divide by }-2 \right)
\\\\&=
\dfrac{\sqrt{3}-\sqrt{7}}{2}
.\end{align*}
Hence, the rationalized-denominator form of the given expression is $
\dfrac{\sqrt{3}-\sqrt{7}}{2}
$.
