The product of the four values of $(1+i \sqrt{3})^{3 / 4}$ is

The product of the four values of $(1+i \sqrt{3})^{3 / 4}$ is
  1. $-8 \mathrm{i}$
  2. i
  3. $-8$
  4. $8$

Solution

Let $z=(1+i \sqrt{3})^{3 / 4}$ Product of four values of $z=|z|^4...(i)$ Now, $z=2^{3 / 4}\left(\frac{1}{2}+i \frac{\sqrt{3}}{2}\right)^{3 / 4}$ $\begin{aligned} & z=2^{3 / 4}\left(e^{\left.i \frac{\pi}{3}\right)^{3 / 4}}=2^{3 / 4} e^{i \frac{\pi}{4}}\right. \\ & \Rightarrow|z|=2^{3 / 4}\left|e^{i \frac{\pi}{4}}\right| \\ & \Rightarrow|z|=2^{3 / 4} \times 1=2^{3 / 4} \\ & \Rightarrow|z|^4=2^3=8 \end{aligned}$ $\therefore \quad$ Product of four values of $z=8$.

Asked in: AP EAMCET 2023 (15 May Shift 2)

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