Question

How do I solve problems like this alt=" {i}^{13} " align="absmiddle" class="latex-formula">

Answer 1

$i^{13}=(\sqrt[2]{-1})^{13}=((-1)^{\frac{1}{2}})^{13}=(-1)^{\frac{13}{2}}=((-1)^{13})^{\frac{1}{2}}=(-1)^{\frac{1}{2}}=\sqrt[]{-1}=i$$i^4=(\sqrt[]{-1})^4=(-1)^{\frac{4}{2}}=(-1)^2=1$

when we power the number i to a odd number, we obtain the same i, but when we power i to a even number we obtain 1 instead.