Question

The value of ${\left(\frac{-1+i\sqrt{3}}{1-i}\right)}^{30}$ is:

• A. $2^{15}i$
• B. $-2^{15}$
• C. $-2^{15}i$
• D. $6^5$

Answer 1

The correct option is C $-2^{15}i$

${\left(\frac{-1+i\sqrt{3}}{1-i}\right)}^{30}$
$={\left[\left(\frac{2\omega }{1-i}\right)\right]}^{30}$
$=\frac{2^{30}{\omega }^{30}}{\left(\right(1-i{)}^2{)}^{15}}$
$=\frac{2^{30}\cdot 1}{(1+i^2-2i{)}^{15}}$
$=\frac{2^{30}}{(-2{)}^{15}{i}^{15}}$
$=\frac{-2^{15}}{-i}$
$=\frac{2^{15}i}{i^2}$
$=-2^{15}i$

Alternate Solution:
${\left(\frac{-1+i\sqrt{3}}{1-i}\right)}^{30}$
$={\left[\left(\frac{-1+i\sqrt{3}}{2}\right)\right(1+i\left)\right]}^{30}$
$=w^{30}(1+i{)}^{30}=(1+2i-1{)}^{15}$
$=-2^{15}i$