Question

(raiz 2 - raiz 6 i) ^ -6

Answer 1

$E=\left(\sqrt{2}-\sqrt{6}i\right)^{-6}\\ \\ E=\left[2\sqrt{2}\times\dfrac{1}{2\sqrt{2}}\times\left(\sqrt{2}-\sqrt{6}i\right)\right]^{-6}\\ \\ \\ E=\left[2\sqrt{2}\times\left(\dfrac{1}{2}-\dfrac{\sqrt{3}}{2}i\right)\right]^{-6}\\ \\ \\ E=\left( 2\sqrt{2}\right)^{-6}[\cos(-60\°)+i\sin(-60\°)]^{-6}\\ \\ \\ E=\dfrac{1}{8^3}[\text{cis}(-60\°)]^{-6}\\ \\ \\ E=\dfrac{1}{8^3}[\text{cis}(-60\°\times -6)]\\ \\ \\ E=\dfrac{1}{8^3}[\text{cis}(-360\°)]$


$E=\dfrac{1}{8^3}[\cos(-360\°)+i\sin(-360\°)]\\ \\ \\ E=\dfrac{1}{8^3}(1+i\cdot 0)\\ \\ \\ E=\dfrac{1}{8^3}\\ \\ \\ \boxed{\boxed{E=\dfrac{1}{512}}}$