Simplify $\frac {\sqrt5}{\sqrt3+1} - \sqrt\frac{30}{8} + \frac {\sqrt {45}}{2}$ I am trying to find the value of: $$\frac {\sqrt5}{\sqrt3+1} - \sqrt\frac{30}{8} +  \frac {\sqrt {45}}{2}$$
I have the key with the answer $\sqrt 5$ but am wondering how I can easily get to that answer?
I realize this is a very basic question on radicals and I know that square roots can be separated into parts, for instance $\sqrt {45} = \sqrt 9 \cdot \sqrt 5$ and I can see how that would be useful here in canceling out radicals but I have not yet been able to reach a solution using this method.

$$\begin{align}
\frac{\sqrt{5}}{\sqrt{3}+1} - \sqrt{\frac{30}{8}} + \frac{\sqrt{45}}{2} &= \frac{\sqrt{5}(\sqrt{3}-1)}{(\sqrt{3}+1)((\sqrt{3}-1))} - \sqrt{\frac{15}{4}} + \frac{\sqrt{9·5}}{2}\\
&= \frac{\sqrt{5}(\sqrt{3}-1)}{2} - \frac{\sqrt{15}}{2} + \frac{3\sqrt{5}}{2}\\ 
&= \frac{\sqrt{15}-\sqrt{5} -\sqrt{15} + 3\sqrt{5}}{2}\\ 
&= \sqrt{5}
\end{align}$$
