First slide

Algebra of complex numbers

Question

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

Easy

A

$- 2^{15}$
B

$- 2^{15} i$
C

$2^{15} i$
D

$6^{5}$

Solution

$we know that - 1 + i \sqrt{3} = 2 ω where ω is cube root of unity$
Hence,
$\begin{matrix} {(- 1 + i \sqrt{3})}^{30} = 2^{30} ω^{30} \\ = 2^{30} \end{matrix}$ $(\sin c e ω^{3} = 1)$
And
$\begin{matrix} (1 - i)^{2} = 1 + i^{2} - 2 i \\ = - 2 i \\ Hence, (1 - i)^{30} = (- 2 i)^{15} = - 2^{15} (- i) = 2^{15} i \\ Therefore, {(\frac{- 1 + i \sqrt{3}}{1 - i})}^{30} = \frac{2^{30}}{2^{15} i} = 2^{15} (- i) \end{matrix} (\sin c e i^{15} = {(i^{4})}^{3} i^{3} = - i)$

Get Instant Solutions

When in doubt download our app. Now available Google Play Store- Doubts App

Recieve an sms with download link

Doubts App

OR

SCAN ME

Doubts App

Doubts App