Explanation

(c)
$\sqrt{- 7 - 24\sqrt{- 1}} = \sqrt{- 1}\sqrt{7 + 24i}$
We know
$\sqrt{a + ib} = \pm \left\lbrack \sqrt{\frac{1}{2}(\sqrt{a^{2} + b^{2}} + a)} + i\sqrt{\frac{1}{2}(\sqrt{a^{2} + b^{2}} - a)} \right\rbrack$
$\therefore i\sqrt{7 + 24i}$
$= i\left\lbrack \pm \left\{ \sqrt{\frac{1}{2}(\sqrt{49 + 576} + 7)} + i\sqrt{\frac{1}{2}(\sqrt{49 + 576} - 7)} \right\} \right\rbrack$
$= i\left\lbrack \pm \left\{ \sqrt{\frac{1}{2}(32)} + i\sqrt{\frac{1}{2}(18)} \right\} \right\rbrack$
$= \pm (3 - 4\sqrt{- 1}$)