Question

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

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

Answer 1

Answer: (A)

$z=\frac{7-i}{3-4i}=\frac{7-i}{3-4i}\times\frac{3+4i}{3+4i}$
$=\frac{21+4+i\left(28-3\right)}{25}=1+i$
$\therefore \,\left|z\right|=\left|1+i\right|=\sqrt{2}$
$\therefore \,\left|z\right|^{14}=\left(\sqrt{2}\right)^{14}$
$=\left[\left(\sqrt{2}\right)^{2}\right]^{7}$
$=2^{7}$