Answer
$-i$
Work Step by Step
Apply exponent rule: $\displaystyle \quad a^{-n}=\frac{1}{a^{n}}$
$i^{-13}=\displaystyle \frac{1}{i^{13}}$
Calculate $i^{13}$:
$i^{13}=i^{12+1}=i^{12}i \qquad$...Apply: $a^{mn}=(a^{m})^{n}$
$=i(i^{2})^{6}\qquad$...Apply: $\quad i^{2}=-1$
$=(-1)^{6}i \qquad$...Apply: $(-a)^{n}=a^{n},$ if $n$ is even
$=1i$
$=i$
Thus,
$i^{-13}=\displaystyle \frac{1}{i}$
Multiply by the conjugate $\displaystyle \frac{-i}{-i}$
$=\displaystyle \frac{1\cdot(-i)}{i(-i)}$
$=1\cdot(-i)$
$=-i$
