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?

If we take the definition of $\pi$ to be based on the circumference of a circle (namely that $\frac{C}{d} = \pi$), then we can see a nice geometric proof of this fact.
Consider a square with side length $1$. Now, draw a circle around this square such that all four corners of the square fall on the circle. This circle has a diameter of $\sqrt{2}$. The perimeter of the square involved is $4$.  
$\frac{4}{\sqrt{2}} = 2^{\frac{3}{2}}$
Now, by looking at this picture it is quite obvious that the circumference of the circle is larger than the perimeter of the square. If you were constructing these shapes this could be easily verified. But now we can say that:
\begin{align}
C &> P_{square} \\
C &> 4 \\
\frac{C}{d} &> \frac{4}{d} \ \ \ \ \textbf{Since $d > 0$} \\
\pi &> 2^{\frac{3}{2}}
\end{align}
