Question

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

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

Answer 1

The correct option is B $256$ and $\frac{2\pi }{3}$

Let $z={(1+i\sqrt{3})}^8$
$={(-2)}^8{\left(\frac{1+i\sqrt{3}}{-2}\right)}^8$
$={(-2)}^8{\left({\omega }^2\right)}^8$
$=2^8{\omega }^{16}$
$=2^8\omega$ $(\because {\omega }^3=1)$
$=2^8\left(\frac{-1+i\sqrt{3}}{2}\right)$
$=2^8(\cos \frac{2\pi }{3}+i\sin \frac{2\pi }{3})$
Therefore, modulus $=2^8=256$ and amplitude $=\frac{2\pi }{3}$