How to simplify an expression

Question

I have an expression:

$$\frac{\sqrt{6} + 1}{6} - \frac{\sqrt{10-4\sqrt{6}}}{6}$$

and it seems like it must be equal to $\frac{1}{2}$. How could i simplify this?

Answer 1

Hint: $(\sqrt{6}-2)^2=10-4\sqrt{6}$.

Answer 2

It is : $(\sqrt{6}-2)^2=10-4\sqrt{6}$ so you get :

$$\frac{\sqrt{6} + 1}{6} - \frac{\sqrt{10-4\sqrt{6}}}{6} = \frac{\sqrt{6} + 1}{6}-\frac{\sqrt{(\sqrt6 - 2)^2}}{6}= \frac{\sqrt{6} + 1}{6}- \frac{\sqrt6-2}{6} = \frac{3}{6} =\frac{1}{2}$$

Note that : $\sqrt{6}-2 > 0 $ and that's why you remove the root simply enough.