The value of
Question:

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

  1. $2^{15} \mathrm{i}$

  2. $-2^{15}$

  3. $-2^{15} \mathrm{i}$

  4. $6^{5}$


Correct Option: , 3

Solution:

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

$=\frac{2^{30} \cdot \omega^{30}}{\left((1-i)^{2}\right)^{30}}$

$=\frac{2^{30} \cdot 1}{\left(1+i^{2}-2 i\right)^{15}}$

$=\frac{2^{30}}{-2^{15} \cdot \mathrm{i}^{15}}$

$=-2^{15} \mathrm{i}$

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