Simplify these expressions with radical sign 2 My question is 

1) Rationalize the denominator: 
  $$\frac{1}{\sqrt{2}+\sqrt{3}+\sqrt{5}}$$

My answer is:
$$\frac{\sqrt{12}+\sqrt{18}-\sqrt{30}}{18}$$
My question is 

2) $$\frac{1}{\sqrt{2}+\sqrt{3}-\sqrt{5}}+\frac{1}{\sqrt{2}-\sqrt{3}-\sqrt{5}}$$

My answer is: $$\frac{1}{\sqrt{2}}$$
I would also like to know whether my solutions are right.
Thank you, 
 A: *

*Your answer is almost correct.


Multiplying by$$\frac{\sqrt{2}+\sqrt{3}-\sqrt{5}}{\sqrt{2}+\sqrt{3}-\sqrt{5}}$$  and simplifying will give your that answer:
$$\frac{\sqrt{2}+\sqrt{3}-\sqrt{5}}{2 \sqrt 6}=\frac{\sqrt{12}+\sqrt{18}-\sqrt{30}}{12}$$
 2. Your answer is correct.
Multiplying the first fraction by $$\frac{\sqrt{2}+\sqrt{3}+\sqrt{5}}{\sqrt{2}+\sqrt{3}+\sqrt{5}}$$ And the second by $$\frac{\sqrt{2}-\sqrt{3}+\sqrt{5}}{\sqrt{2}-\sqrt{3}+\sqrt{5}}$$
A: \begin{align}
\dfrac1{\sqrt{2} + \sqrt{3} + \sqrt{5}} & = \dfrac{\sqrt{2} + \sqrt3 - \sqrt5}{(\sqrt{2} + \sqrt{3} + \sqrt{5})(\sqrt{2} + \sqrt{3} - \sqrt{5})}\\
& = \dfrac{\sqrt{2} + \sqrt3 - \sqrt5}{(\sqrt{2} + \sqrt{3})^2 - 5}\\
& = \dfrac{\sqrt{2} + \sqrt3 - \sqrt5}{(2+3+2 \sqrt{6}) - 5}\\
& = \dfrac{\sqrt{2} + \sqrt3 - \sqrt5}{2 \sqrt{6}}\\
& = \dfrac{(\sqrt{2} + \sqrt3 - \sqrt5) \sqrt{6}}{2 \times 6}\\
& = \dfrac{(\sqrt{12} + \sqrt{18} - \sqrt{30} )}{12}\\
\end{align}
Hence, your denominator should be $12$ and not $18$ for the first problem.
For the second problem,
\begin{align}
\dfrac1{\sqrt{2} + \sqrt{3} - \sqrt{5}} + \dfrac1{\sqrt{2} - \sqrt{3} - \sqrt{5}} & = \dfrac1{(\sqrt{2} - \sqrt{5}) + \sqrt{3}} + \dfrac1{(\sqrt{2} - \sqrt{5}) - \sqrt{3}} \\
& = \dfrac{2(\sqrt{2} - \sqrt{5})}{(\sqrt{2} - \sqrt{5})^2 - 3}\\
& = \dfrac{2(\sqrt{2} - \sqrt{5})}{7 - 2 \sqrt{10} - 3}\\
& = \dfrac{\sqrt{2}(2 - \sqrt{10})}{4 - 2 \sqrt{10}}\\
& = \dfrac{\sqrt{2}(2 - \sqrt{10})}{2(2 - \sqrt{10})}\\
& = \dfrac{\sqrt{2}}{2}
\end{align}
Hence, your second answer is indeed correct. You may want to rationalize the denominator by multiplying by $\sqrt{2}$.
A: $\begin{eqnarray*}
(\sqrt{2}+\sqrt{3}+\sqrt{5})(\sqrt{12}+\sqrt{18}-\sqrt{30}) & = & (\sqrt{2}+\sqrt{3}+\sqrt{5})(2\sqrt{3}+3\sqrt{2}-\sqrt{2}\sqrt{3}\sqrt{5})\\& = & 12,
\end{eqnarray*}$
if you expand out the terms, so your first answer is incorrect. The denominator should be $12$.
$\begin{eqnarray*}
(\sqrt{2}+\sqrt{3}-\sqrt{5})(\sqrt{2}-\sqrt{3}-\sqrt{5}) & = & (\sqrt{2}-\sqrt{5})^2-\sqrt{3}^2\\& = & 7-2\sqrt{10}-3\\& = & 2\sqrt{2}(\sqrt{2}-\sqrt{5}),
\end{eqnarray*}$
and so when your fractions in the second part are given common denominators, you'll have exactly $\cfrac{1}{\sqrt{2}}$ after cancellation, so your second answer is correct.
Note: In general, if you want to see if two fractions are the same (as in the first problem), cross-multiplication is often a useful way to see it.
