Question

If $z$ is a nonreal root of $\sqrt[7]{7-1}$, then the value of $z^{86}+z^{175}+z^{289}$

• A. $-i$
• B. $i$
• C. $-1$
• D. $1$

Answer 1

The correct option is C $-1$

Let $z=\sqrt[7]{7-1}$. Then
$z^7=-1$
$\therefore z^{86}+z^{175}+z^{289}$
$=(z^7{)}^{12}\times z^2+(z^7{)}^{25}+(z^7{)}^{41}\times z^2$
$=(-1{)}^{12}\times z^2+(-1{)}^{25}+(-1{)}^{41}\times z^2$
$=z^2-1-z^2=-1$

Hence, option C.