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

JEE Main · 2020 · 05 Sep 2020 · Shift 2 · Complex Numbers

The value of

(\frac{- 1 + i 3}{1 - i})^{30}

is:

Answer: (D)

Write the parts in polar form:

- 1 + i 3 = 2 e^{i 2 π /3}

and

1 - i = 2 e^{- iπ /4}

. The base is

\frac{2}{2} e^{i (2 π /3 + π /4)} = 2 e^{i 11 π /12}

. Raising to the thirtieth power, the modulus becomes

(2)^{30} = 2^{15}

and the argument becomes

30 \times \frac{11 π}{12} = \frac{55 π}{2}

, which is coterminal with

\frac{3 π}{2}

. Since

e^{i 3 π /2} = - i

, the value is

2^{15} (- i) = - 2^{15} i

JEE Main · 2020 · 05 Sep 2020 · Shift 2 · Single correct option