ANSWER
[tex] {i}^{41}=i[/tex]
EXPLANATION
We want to simplify,
[tex] {i}^{41} [/tex]
We rewrite the exponent to get,
[tex] {i}^{40 + 1} [/tex]
We apply the laws of indices to get,
[tex] {i}^{40} \times {i}^{1} [/tex]
We apply the laws of indices again to get,
[tex]( {i}^{2} )^{20} \times i[/tex]
Recall that,
[tex]{i}^{2} = - 1[/tex]
[tex] = ( - 1)^{20} \times i[/tex]
The exponent is even so we obtain a positive result
[tex] = 1\times i[/tex]
This gives us,
[tex] = i[/tex]
