Write [tex]1+2i[/tex] in polar form [tex]re^{i\theta}[/tex]. We don't care what the argument [tex]\theta[/tex] is, so we'll ignore it. Meanwhile,
[tex]r=\sqrt{1^2+2^2}=\sqrt5[/tex]
and by DeMoivre's theorem,
[tex](re^{i\theta})^4=r^4e^{4i\theta}[/tex]
i.e. the modulus of [tex](1+2i)^4[/tex] is simply [tex](\sqrt5)^4=(5^{1/2})^4=5^2=25[/tex].
