Answer
$-i$
Work Step by Step
In order to calculate the larger powers of $i$, we can first calculate the values of $i$, $i^{2}$, $i^{3}$, and $i^{4}$.
$i^{1}=i$
$i^{2}=i\times i=-1$
$i^{3}=i\times i^{2}=i\times-1=-i$
$i^{4}=i^{2}\times i^{2}=-1\times-1=1$
We can use these values and the rules of exponents to derive larger powers of $i$.
Therefore, $i^{-5}=\frac{1}{i^{5}}=\frac{1}{i^{4}\times i^{1}}=\frac{1}{1\times i}=\frac{1}{i}=\frac{1(-i)}{i(-i)}=\frac{-i}{-i^{2}}=\frac{-i}{-(-1)}=\frac{-i}{1}=-i$.
