Answer
$\vert nm \vert = \vert n \vert~\vert m \vert$
Work Step by Step
$n = a + ib$
$m = c + id$
We can find an expression for $\vert nm \vert$:
$\vert nm \vert = \vert (a + ib)(c + id) \vert$
$\vert nm \vert = \vert (ac-bd) + i(bc + ad) \vert$
$\vert nm \vert = \sqrt{(ac-bd)^2 + (bc + ad)^2}$
$\vert nm \vert = \sqrt{(ac)^2-2abcd+ (bd)^2 + (bc)^2 + 2abcd+(ad)^2}$
$\vert nm \vert = \sqrt{(ac)^2+ (bd)^2 + (bc)^2+(ad)^2}$
$\vert nm \vert = \sqrt{(a^2+ b^2)(c^2+ d^2)}$
$\vert nm \vert = \sqrt{(a^2+ b^2)}~\sqrt{(c^2+ d^2)}$
$\vert nm \vert = \vert n \vert~\vert m \vert$
