How to simplify a square root How can the following:
$$
\sqrt{27-10\sqrt{2}}
$$
Be simplified to:
$$
5 - \sqrt{2}
$$
Thanks

If you're faced with a question that says "Prove that $\sqrt{27-10\sqrt{2}}$ $=5 - \sqrt{2}$", then it's just a matter of squaring $5 - \sqrt{2}$ and seeing that you get $27-10\sqrt{2}$.  But suppose the question your faced with is to find a square root of $27-10\sqrt{2}$ of the form $a+b\sqrt{2}$, where $a$ and $b$ are rational.  Then you have
$$
27-10\sqrt{2}=\left(a+b\sqrt{2}\right)^2 = a^2 + 2ab\sqrt{2} + 2b^2
$$
so
\begin{align}
27 & = a^2+2b^2 \\[8pt]
-10 & = 2ab
\end{align}
From the second equation we get $a=-5/b$, then the first equation becomes
$$
27 = \frac{25}{b^2} + 2b^2
$$
or
$$
2(b^2)^2 -27b^2 + 25 = 0.
$$
A solution is $b^2=1$, and you can go on from there to find $b$ and then $a$.
(And remember that the number will have two square roots.)
Later note: In order for all this to work, we have to rely on the fact that $\sqrt{2}$ is irrational.  That enables us to conclude that the rational parts are equal and the irrational parts are equal, so we have two equations.
