Q47 / 160 · AP EAMCET 2022 (06 Jul Shift 2) · Mathematics

$$ (-1+i \sqrt{3})^{60}= $$

  • Correct answer
    A
    $2^{60}$
  • B
    $2^{59}$
  • C
    $2^{61}$
  • D
    $2^{30}$

Solution

Here, $(-1+i \sqrt{3})^{60}$ $$ \begin{array}{lrl} & =2^{60}\left[-\frac{1}{2}+i \frac{\sqrt{3}}{2}\right]^{60} & \\ & =2^{60} \times \omega^{60} & {\left[\because \omega=-\frac{1}{2}+i \frac{\sqrt{3}}{2}\right]} \\ & =2^{60} \times\left(\omega^3\right)^{20} & {\left[\because \omega^3=1\right]} \\ & =2^{60} & {\left[\because \omega^6\right.} \end{array} $$

Hence, the correct answer is option: 'A'.

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