Cho z=(1+i√3/√3+i)^120. Tính |z|.

Question

Cho \(z=\left(\frac{1+i\sqrt{3} }{\sqrt{3} +i} \right)^{120}\) . Tính \(\left|z\right|.\)

A. 2.

B. 1.

C. 3.

D. 4.

Answer 1

Chọn B

Ta có: \(z_{1} =\frac{1+i\sqrt{3} }{\sqrt{3} +i} =\frac{\sqrt{3} }{2} +\frac{1}{2} i\)
\(\Rightarrow z_{1}^{3} =\left(\frac{\sqrt{3} }{2} +\frac{1}{2} i\right)^{3} =i.\)
Suy ra \(z=z_{1}^{120} =\left(z_{1}^{3} \right)^{40} =i^{40} =\left(i^{4} \right)^{10} =1.\)

Vậy \(\left|z\right|=1.\)