Question

What is the simplest form of the expression? Please show work

Answer 1

We need to rationalize the denomenator

remember
(a+b)(a-b)=a²-b²
the bottom is the (a+b) form


mutliply it by $\frac{ \sqrt{2}- \sqrt{10} }{\sqrt{2}- \sqrt{10}}$
$(\frac{ \sqrt{2}- \sqrt{10} }{\sqrt{2}+ \sqrt{10}})(\frac{ \sqrt{2}- \sqrt{10} }{\sqrt{2}- \sqrt{10}})$=
$\frac{ (\sqrt{2}- \sqrt{10})^2 }{(\sqrt{2})^2- (\sqrt{10})^2}$=
$\frac{ 12-4\sqrt{5} }{2-10}$=
$\frac{ 12-4\sqrt{5} }{-8}$=
$\frac{ \sqrt{5}-3 }{2}$