ANSWER
[tex] {i}^{31} = - i[/tex]
EXPLANATION
We want to evaluate
[tex] {i}^{31} [/tex]
Use indices to rewrite the expression as:
[tex] = {i}^{30} \times i[/tex]
We know that
[tex] {i}^{2} = - 1[/tex]
So we rewrite the expression to obtain;
[tex] = ({ {i}^{2}) }^{15} \times i[/tex]
This gives us;
[tex]= {( - 1) }^{15} \times i[/tex]
This simplifies to
[tex] = - 1 \times i[/tex]
[tex] = - i[/tex]
