# How do you simplify  i^5?

Feb 10, 2016

This is the same as i

#### Explanation:

The complex i repeats after 4 powers

This can be seen with
i^1 = i
i^2 = ${\sqrt{- 1}}^{2}$ = -1
i^3 = i^2 * i = -1 * i => -i
i^4 = i^2 * i^2 => -1 * -1 >>> 1

After this, we can take the mod 4 of the exponent
(in this case we can just do the next exponent, try it out and of course the answer will be the same as i ^ 1 )

Feb 10, 2016

$\text{simplified "->" } i$

#### Explanation:

Write as $\text{ } {i}^{2} \times {i}^{2} \times i$

But ${i}^{2} = - 1 \text{ }$so we have:

$\left(- 1\right) \times \left(- 1\right) \times i \text{ "=" } i$