Answer
$-2+2\sqrt{3}$
Work Step by Step
Multiplying the numerator and the denominator by the conjugate of the denominator, the given expression, $
\dfrac{4}{1+\sqrt{3}}
,$ is equivalent to
\begin{align*}\require{cancel}
&
\dfrac{4}{1+\sqrt{3}}\cdot\dfrac{1-\sqrt{3}}{1-\sqrt{3}}
\\\\&=
\dfrac{4(1-\sqrt{3})}{1^2-(\sqrt{3})^2}
&\left( \text{use }(a+b)(a-b)=a^2-b^2 \right)
\\\\&=
\dfrac{4-4\sqrt{3}}{1^2-(\sqrt{3})^2}
&\left( \text{use Distributive Property} \right)
\\\\&=
\dfrac{4-4\sqrt{3}}{1-3}
\\\\&=
\dfrac{4-4\sqrt{3}}{-2}
\\\\&=
\dfrac{\cancel4^{-2}\cancel{-4}^{2}\sqrt{3}}{\cancel{-2}^1}
&\left( \text{divide by }-2 \right)
\\\\&=
-2+2\sqrt{3}
.\end{align*}
Hence, the rationalized-denominator form of the given expression is $
-2+2\sqrt{3}
$.