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)

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