Question

What is the solution i^401?

Answer 1

There is a set of rules to solve i to the power.
First off, you need to know this:
${i}^{1} = i \\ i^{2} = - 1 \\ {i}^{3} = - i \\ {i}^{4} = 1$
After the power of 4, it just starts to loop from 1. Knowing this, we can say the following:
$i ^{4k} = 1 \\ {i}^{4k + 1} = i \\ {i}^{4k + 2} = - 1 \\ {i}^{4k + 3} = - i$
Applying this, all we need to do is divide the power by 4 and determine the remainder.
$i ^{401} = \\ 401 \times 4 = 100 \: (1 \: left) \\ i^{401} = i^{4 \times 100 + 1} \\ k = 100 \\ {i}^{401} = i^{4 \times 100 + 1} = {i}^{4k + 1} = i$
So the answer is:
${i}^{401} = i$