Answer
$ \frac{5(3-\sqrt{5})}{2}= 7.5-2.5\sqrt{5}$
Work Step by Step
Given \begin{equation}
\frac{10}{3+\sqrt{5}}.
\end{equation} Rationalize the denominator and simplify.
\begin{equation}
\begin{aligned}
\frac{10}{3+\sqrt{5}}& =\frac{10}{3+\sqrt{5}}\cdot\left(\frac{3-\sqrt{5}}{3-\sqrt{5}}\right) \\
& =\frac{10(3-\sqrt{5})}{9-5} \\
& =\frac{10(3-\sqrt{5})}{4} \\
& =\frac{5(3-\sqrt{5})}{2} \\
&= 7.5-2.5\sqrt{5}.
\end{aligned}
\end{equation} We got
\begin{equation}
\frac{10}{3+\sqrt{5}}= \frac{5(3-\sqrt{5})}{2}= 7.5-2.5\sqrt{5}.
\end{equation}
