How do you simplify $i^{100}$ $i^100$ ?

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Answer 1 · 2018-04-25T17:51:17

$i^{100} = 1$ $i^100=1$

$i^{100} = {(i^{2})}^{50}$ $i^100=(i^2)^50$

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

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

Alternatively, we can rewrite in trigonometric form and then in the form $r e^{i θ}$ $re^(itheta)$ :

$i = cos (\frac{π}{2}) + i sin (\frac{π}{2})$ $i=cos(pi/2)+isin(pi/2)$

$= e^{i \frac{π}{2}}$ $=e^(ipi/2)$

Raise the exponential to the power of $100 :$ $100:$

${(e^{i \frac{π}{2}})}^{100} = e^{50 π}$ $(e^(ipi/2))^100=e^(50pi)$

$= cos (50 π) + i sin (50 π)$ $=cos(50pi)+isin(50pi)$

$= cos 2 π + i sin 2 π$ $=cos2pi+isin2pi$

$cos 2 π = 1, sin 2 π = 0$ $cos2pi=1, sin2pi=0$

so we get

$= 1$ $=1$

Answer 2 · 2018-04-25T17:55:06

$i^{100} = 1$ $i^100=1$

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

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

How do you simplify i^100i100i^100?