Q58 / 120 · VITEEE 2017 · Mathematics

If $z=\frac{7-i}{3-4 i}$ then $z^{14}=$

  • A
    $2^{7}$
  • B
    $2^{7} i$
  • C
    $2^{14} i$
  • Correct answer
    D
    $-2^{7} i$

Solution

$\begin{aligned} z=\frac{7-i}{3-4 i} \times \frac{3+4 i}{3+4 i} & \\=\frac{21+25 i+4}{16+9}=\frac{25(1+i)}{25}=(1+i) \\ z^{14}=(1+i)^{14}=\left[(1+i)^{2}\right]^{7}=(2 i)^{7} \\=2^{7} i^{7}=-2^{7} i \end{aligned}$

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

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