Question

O número complexo (1 + i)²² é:Sua primeira pergunta? *

Answer 1

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Potênciação de números complexos na forma trigonométrica

$\Large\boxed{\boxed{\boxed{\boxed{\sf{Z^n=\rho^n\cdot\left[cos(n\theta)+i\,sen(n\theta)\right]}}}}}$

$\sf{z=1+i}\\\sf{\rho=\sqrt{1^2+1^2}=\sqrt{2}}\\\sf{cos(\theta)=\dfrac{1}{\sqrt{2}} =\dfrac{\sqrt{2}}{2}}\\\sf{sen(\theta)=\dfrac{1}{\sqrt{2}}=\dfrac{\sqrt{2}}{2}}\\\sf{\theta=\dfrac{\pi}{4}}$

$\sf{(1+i)^{22}=(\sqrt{2})^{22}\cdot\left[cos\left(22\cdot\dfrac{\pi}{4}\right)+i\,sen\left(22\cdot\dfrac{\pi}{4}\right)\right]}\\\sf{(1+i)^{22}=2048\left[cos\left(\dfrac{11\pi}{2}\right)+i\,sen\left(\dfrac{11\pi}{2}\right)\right]}\\\sf{(1+i)^{22}=2048\cdot\left[0+i\cdot(-1)\right]}$

$\huge\boxed{\boxed{\boxed{\boxed{\sf{(1+i)^{22}=-2048i}}}}}$