# Problems from Exponentiation of the imaginary unit

Find the following powers of the imaginary unit:

• $$i^{117}$$
• $$i^{43}$$
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### 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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Compute the following values:

• $$(4i)^3$$
• $$5i^{16}-81$$
• $$\dfrac{i^{24}}{i^{11}}$$
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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: $$5i^{16}-81=5\cdot i^0-81=5\cdot1-81=5-81=-76$$$
• 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: $$i^{13}=i^1=i$$$

### Solution:

• $$-64i$$
• $$-76$$
• $$i$$
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