Problems from Exponentiation of the imaginary unit

Development:

• 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, $i^{43}=i^3=-1$.

Solution:

• $i$
• $-i$

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Development:

• 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: $$5i^{16}-81=5\cdot i^0-81=5\cdot 1-81=5-81=-76$$
• Recalling that the division of two powers (sharing the same base) is the difference of their powers we have that, $${\displaystyle \frac{i^{24}}{i^{11}}}=i^{24-11}=i^{13}$$ Then, dividing $13$ by $4$ we obtain a reminder of $1$. Thus: $$i^{13}=i^1=i$$

Solution:

• $-64i$
• $-76$
• $i$

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