Problems from Exponentiation of the imaginary unit

Find the following powers of the imaginary unit:

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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, $$i^{43}=i^3=-1$$.

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Compute the following values:

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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: $$$i^{13}=i^1=i$$$

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