Answer
$13+7\sqrt{3}$
Work Step by Step
Multiplying the numerator and the denominator by the conjugate of the denominator, the given expression, $
\dfrac{5+\sqrt{3}}{2-\sqrt{3}}
,$ is equivalent to
\begin{align*}\require{cancel}
&
\dfrac{5+\sqrt{3}}{2-\sqrt{3}}\cdot\dfrac{2+\sqrt{3}}{2+\sqrt{3}}
\\\\&=
\dfrac{(5+\sqrt{3})(2+\sqrt{3})}{(2)^2-(\sqrt{3})^2}
&\left( \text{use }(a+b)(a-b)=a^2-b^2 \right)
\\\\&=
\dfrac{5(2)+5(\sqrt{3})+\sqrt{3}(2)+\sqrt{3}(\sqrt{3})}{(2)^2-(\sqrt{3})^2}
&\left( \text{use FOIL} \right)
\\\\&=
\dfrac{10+5\sqrt{3}+2\sqrt{3}+3}{4-3}
\\\\&=
\dfrac{13+7\sqrt{3}}{1}
\\\\&=
13+7\sqrt{3}
.\end{align*}
Hence, the rationalized-denominator form of the given expression is $
13+7\sqrt{3}
$.
