# How do you simplify i^752?

##### 1 Answer
Jan 24, 2016

$1$

#### Explanation:

Recall that $i = \sqrt{- 1}$. Because of this, ${i}^{2} = - 1$. We can then use this fact to find ${i}^{4}$:

${i}^{2} = - 1$
${\left({i}^{2}\right)}^{2} = {\left(- 1\right)}^{2}$
${i}^{4} = 1$

The fact that ${i}^{4} = 1$ is very useful. In fact, ${i}^{752}$ can be expressed solely in terms of ${i}^{4}$.

Since $188 \times 4 = 752$, we can say that

${i}^{752} = {\left({i}^{4}\right)}^{188}$

And ${i}^{4} = 1$, so

${i}^{724} = {1}^{188} = 1$