# How do you simplify i^100?

Apr 25, 2018

${i}^{100} = 1$

#### Explanation:

${i}^{100} = {\left({i}^{2}\right)}^{50}$

From the fact that ${i}^{2} = - 1 ,$ we get

${\left(- 1\right)}^{50} = 1$ as $- 1$ raised to any even power is $1.$

Alternatively, we can rewrite in trigonometric form and then in the form $r {e}^{i \theta}$:

$i = \cos \left(\frac{\pi}{2}\right) + i \sin \left(\frac{\pi}{2}\right)$

$= {e}^{i \frac{\pi}{2}}$

Raise the exponential to the power of $100 :$

${\left({e}^{i \frac{\pi}{2}}\right)}^{100} = {e}^{50 \pi}$

$= \cos \left(50 \pi\right) + i \sin \left(50 \pi\right)$

$= \cos 2 \pi + i \sin 2 \pi$

$\cos 2 \pi = 1 , \sin 2 \pi = 0$

so we get

$= 1$

Apr 25, 2018

${i}^{100} = 1$

#### Explanation:

${i}^{100} = {\left({i}^{2}\right)}^{50} = {\left(- 1\right)}^{50} = 1$

${\left(- a\right)}^{n} = {a}^{n}$, where n is an even number.