Answer
$-2\sqrt{6} - 2\sqrt{10}i$
Work Step by Step
RECALL:
(1) $\sqrt{a} \cdot \sqrt{b} = \sqrt{ab} \text{ where } a, b, \gt 0$
(2) $a(b+c) =ab+ac$
(3) $i^2=-1$
(4) For any real number $a \gt 0$, $\sqrt{-a} = i\sqrt{a}$
Use rule (4) above to obtain:
$=i\sqrt{8}(i\sqrt{3}-\sqrt5)
\\=i\sqrt{4(2)}(i\sqrt{3}-\sqrt{5})
\\=i\sqrt{2^2(2)}(i\sqrt{3}-\sqrt{5})
\\=i\cdot 2\sqrt{2}(i\sqrt{3}-\sqrt{5})
\\=2i\sqrt{2}(i\sqrt{3}-\sqrt{5})$
Use rules (1) and (2) to obtain:
$=2i^2\sqrt{6} - 2\sqrt{10}i$
Use rule (3) above to obtain:
$=2(-1)\sqrt{6} - 2\sqrt{10}i
\\=-2\sqrt{6} - 2\sqrt{10}i$
