Example

Write $i^5$ as a complex number in standard form.

Solution

Powers of $i$ follow a pattern:

Here is an easy way to use this pattern for simplifying higher powers of $i$:

To simplify $i^p$...

1. Find the remainder when $p$ is divided by 4. Call it $r$, and notice that $r=0$, $1$, $2$, or $3$.
2. Then $i^p=i^r$.

For example, to simplify $i^{27}$, we first notice that $27 \div 4 = 6 \mbox{ r } 3$. It follows that