How to prove: $2^\frac{3}{2}<\pi$ without writing the explicit values of $\sqrt{2}$ and $\pi$ How to prove: $2^\frac{3}{2}<\pi$  without writing the explicit values of $\sqrt{2}$ and $\pi$.
I am trying by calculus but don't know how to use here in this problem. Any idea?

It's pretty easy to see that $\pi > 3$, by inscribing a circle inside a regular hexagon. Then squaring both sides gives $\pi^2>9>8$. Taking square roots again gives $\pi > \sqrt{8}=2^\frac{3}{2}$
