Exponentiation of the imaginary unit

Find the following powers of the imaginary unit:

In this case, $117$ is divided by $4$ and we obtain a reminder of $1$. So, $i^{117}=i^1=i$.
In this case, $43$ is divided by $4$ and we obtain a reminder of $3$. So, $i^{43}=i^3=-1$.

Compute the following values:

For these calculation we will use the power of $i$ that we have learned. $$ (4i)^3=4^3\cdot i^3=64\cdot(-i)=-64i$$
In this case, we obtain that the reminder of $16$ divided by $4$ is $0$, since: $$ 5i^{16}-81=5\cdot i^0-81=5\cdot1-81=5-81=-76$$
Recalling that the division of two powers (sharing the same base) is the difference of their powers we have that, $$\dfrac{i^{24}}{i^{11}}=i^{24-11}=i^{13}$$ Then, dividing $13$ by $4$ we obtain a reminder of $1$. Thus: $$i^{13}=i^1=i$$