# Thread: Which side is larger?

1. ## Which side is larger?

Determine which side is larger without using a calculator
or computer with showing your steps.

$81^{81.25} \ \ vs. \ \ 82^{81}$

2. ## Re: Which side is larger?

Spoiler:

$81^{81.25} = 3\cdot 81^{81}$

$82^{81} =81^{81} \left(\dfrac{1+81}{81}\right)^{81}$

$\lim_{n \to \infty} ~\left(\dfrac{1+n}{n}\right)^n = e$

and this limit is approached strictly from below.

Thus $\left(\dfrac{1+81}{81}\right)^{81} < e < 3$

So comparing the two expressions we have

$3 \cdot 81^{81} > \left(\dfrac{1+81}{81}\right)^{81} \cdot 81^{81}$

and thus $81^{81.25} > 82^{81}$