Question

If $Z=\frac{7+i}{3+4i}$, then value of $z^{14}$ is

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

Answer 1

The correct option is C $2^{7}i$

$Z=\frac{7+i}{3+4i}=\frac{(7+i)(3-4i)}{25}=\frac{25-25i}{25}=1-i\therefore z^{14}=(z^2{)}^7=[(1-i{)}^2{]}^7=(-2i{)}^7=(-2{)}^7=-2^7(-i)$