Question

If $z=\frac{7-i}{3-4i},$ then $z^{14}$ is equal to:

• A. $2^7$
• B. $2^{7}i$
• C. $2^{14}i$
• D. $-2^{7}i$
• E. $-2^{14}$

Answer 1

Answer: (D)

$z=\frac{7-i}{3-4i}=\frac{(7-i)(3+4i)}{{(3)}^2-{(4i)}^2}$
$=(1+i)$
$\therefore$ $z^{14}={(1+i)}^{14}$
$={\left\{\sqrt{2}\left(\cos \frac{\pi }{4}+i\sin \frac{\pi }{4}\right)\right\}}^{14}$
$=2^7\left(\cos \frac{7\pi }{2}+i\sin \frac{7\pi }{2}\right)$
$=2^7(-i)=-2^{7}i$