A Radical Problem

$\space\space\space\space(\sqrt{4 + \sqrt{7}} - \sqrt{4 - \sqrt{7}})^2$

$=(4+\sqrt{7})+(4-\sqrt{7})-2\sqrt{(4+\sqrt{7})(4-\sqrt{7})}$

$=8-2\sqrt{9}$

$=2$

So, $\sqrt{4 + \sqrt{7}} - \sqrt{4 - \sqrt{7}}=\sqrt{2}$

$\begin{aligned} x & = \sqrt{4+\sqrt 7} - \sqrt{4-\sqrt 7} \\ & = \sqrt {\left(\frac {\sqrt 7+1}{\sqrt 2}\right)^2} - \sqrt {\left(\frac {\sqrt 7-1}{\sqrt 2}\right)^2} \\ & = \frac {\sqrt 7+1}{\sqrt 2} - \frac {\sqrt 7-1}{\sqrt 2} \\ & = \frac 2{\sqrt 2} = \boxed{\sqrt 2} \end{aligned}$