Fall 2021 AMC 10 B

Answer (E): Note that $8=2^3$, so $$n=8^{2022}=(2^3{)}^{2022}=2^{6066}=(2^2{)}^{3033}=4^{3033}.$$ Thus $$\frac{n}{4}=\frac{4^{3033}}{4}=4^{3032}.$$

All the other choices are less than $4^{3032}$. Choices (A) and (D) have the same base but a lesser exponent. Choice (B) is incorrect because $2^{2022}<4^{2022}<4^{3032}$. Choice (C) is incorrect because $$8^{2018}=(4^{\frac{3}{2}}{)}^{2018}=4^{3027}<4^{3032}.$$