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I know a much simpler way; you just need to describe for him/her how to exponentiate.

Let $a , b > 2$ (with $a b \ne b$) ge e$ be two numbers (for simplicity, let them be positive integers). The question is, which one is greater: $a^b$, or $b^a$? For instance, let $a=1000$ and $b=999$. Which one is correct: $1000^{999} > 999^{1000}$, or $1000^{999} < 999^{1000}$? (I assume that equality does NOT hold.)

Let the following hold: $|a-e| > |b-e|$. Then, it

It can be shown that $a^b < b^a$. In fact, for every $x \ge 0$, we have $e^x > x^e$ (assuming $x \ne e$).
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I know a much simpler way; you just need to describe for him/her how to exponentiate.

Let $a , b > 2$ (with $a \ne b$) be two numbers (for simplicity, let them be positive integers). The question is, which one is greater: $a^b$, or $b^a$? For instance, let $a=1000$ and $b=999$. Which one is correct: $1000^{999} > 999^{1000}$, or $1000^{999} < 999^{1000}$? (I assume that equality does NOT hold.)

Let the following hold: $|a-e| > |b-e|$. Then, it can be shown that $a^b < b^a$.

In fact, for every $x \ge 0$, we have $e^x > x^e$ (assuming $x \ne e$).
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I know a much simpler way; you just need to describe for him/her how to exponentiate.

Let $a , b > 1$ 2$ be two numbers (for simplicity, let them be positive integers). The question is, which one is greater: $a^b$, or $b^a$? For instance, let $a=1000$ and $b=999$. Which one is correct: $1000^{999} > 999^{1000}$, or $1000^{999} < 999^{1000}$? (I assume that equality does NOT hold.)

Let the following hold: $|a-e| > |b-e|$. Then, it can be shown that $a^b < b^a$.

In fact, for every $x \ge 0$, we have $e^x > x^e$ (assuming $x \ne e$).
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