Question

The modulus and amplitude of ${\left(1+i\sqrt{3}\right)}^8$ are respectively,

• A. $256$ and $\frac{\mathrm{\pi }}{3}$
• B. $256$ and $\frac{2\mathrm{\pi }}{3}$
• C. $2$ and $\frac{2\mathrm{\pi }}{3}$
• D. $256$ and $\frac{8\mathrm{\pi }}{3}$

Answer 1

The correct option is B

$256$ and $\frac{2\mathrm{\pi }}{3}$

Explanation for the correct option:

Find the modulus and amplitude of the given complex number

In the question, a complex number ${\left(1+i\sqrt{3}\right)}^8$ is given.

rewrite the given complex number as follows,

$$\begin{gathered} {\left(1+i\sqrt{3}\right)}^8={\left[2\left(\frac{1}{2}+i\frac{\sqrt{3}}{2}\right)\right]}^8 \\ \Rightarrow {\left(1+i\sqrt{3}\right)}^8={\left[2\left(\sin \left(\frac{\mathrm{\pi }}{3}\right)+i\cos \left(\frac{\mathrm{\pi }}{3}\right)\right)\right]}^8 \end{gathered}$$

According to the Euler's representation,

$$\begin{gathered} {\left(1+i\sqrt{3}\right)}^8={\left[2e^{i\frac{\mathrm{\pi }}{3}}\right]}^8 \\ \Rightarrow {\left(1+i\sqrt{3}\right)}^8={\left(2\right)}^8{e}^{8i\frac{\mathrm{\pi }}{3}} \end{gathered}$$

So, the modulus of the given complex number can be given by, $2^8=256$.

The amplitude of the given complex can be given by, $\frac{2\mathrm{\pi }}{3}$.

Therefore, the modulus and amplitude of the given complex number are $256$ and $\frac{2\mathrm{\pi }}{3}$ respectively.

Hence, the correct option is $B$.