Question

The modulus and amplitude of the complex number $[e^{3-i\frac{\pi }{4}}{]}^3$ are respectively.

• A. $e^9,\frac{\pi }{2}$
• B. $e^9,\frac{-\pi }{2}$
• C. $e^6,\frac{-3\pi }{4}$
• D. $e^9,\frac{-3\pi }{4}$

Answer 1

The correct option is D $e^9,\frac{-3\pi }{4}$

$z=(e^{3-\frac{i\pi }{4}}{)}^3$
$=(e^3.e^{-\frac{i\pi }{4}}{)}^3$
$=e^9.e^{-\frac{3i\pi }{4}}$
$=|z|e^{i\arg \left(z\right)}$
By comparing RHS and LHS we get
$|z|=e^9$ and $\arg \left(z\right)=\frac{-3\pi }{4}$.