<s>[INST] What Is Bigger $100^{100}$or $\sqrt{99^{99} \cdot 101^{101}}$ Hello every what is bigger $100^{100}$or $\sqrt{99^{99} \cdot 101^{101}}$?
I tried to square up and I got $100^{200}$ or $99^{99} \cdot 101^{101}$ and I don't have an idea how to continue.
 [/INST] It is “well-known” that $\log(u) \le u-1$ for all $u>0$, with strict inequality for $u \ne 1$ (see for example for each $x>1 , \frac{x-1}{x}\ < \ln x < x-1$).
It follows that for all real $x > 1$
$$
 \log \left( \frac{x^{2x}}{(x-1)^{x-1} (x+1)^{x+1}}\right)
= (x-1) \log \left( \frac{x}{x-1}\right) + (x+1) \log \left( \frac{x}{x+1}\right) \\
< (x-1) \left( \frac{x}{x-1} - 1 \right) + (x+1) \left( \frac{x}{x+1} - 1\right)
= 1 - 1 = 0
$$
and therefore
$$ 
x^x < \sqrt{(x-1)^{x-1} (x+1)^{x+1}} \, .
$$
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