Tính ( 1+ (căn 3) i) ^ 2021

Đáp án:

$2^{2021}\left(\cos\dfrac{\pi}{3} - i\sin\dfrac{\pi}{3}\right)= 2^{2020} - 2^{2020}\sqrt3 i$

Giải thích các bước giải:

$\begin{array}{l}\quad z = (1 + \sqrt3i)^{2021}\\ Xét\,\,w = 1 + \sqrt3 i\\ \to w = 2\left(\dfrac12 + \dfrac{\sqrt3}{2}i\right)\\ \to w = 2\left(\cos\dfrac{\pi}{3} + i\sin\dfrac{\pi}{3}\right)\\ \text{Ta được:}\\ \quad z = w^{2021}\\ \to z = 2^{2021}\left(\cos\dfrac{2021\pi}{3} + i\sin\dfrac{2021\pi}{3}\right)\quad \text{(Công thức Moivre)}\\ \to z = 2^{2021}\left(\cos\dfrac{\pi}{3} - i\sin\dfrac{\pi}{3}\right)\\ \to z = 2^{2020} - 2^{2020}\sqrt3 i \end{array}$