Which is bigger?

Problem: Which is bigger…\(3^{\pi}\) or \({\pi}^{3}\)?

Solution: Write …\(\pi=3+\delta\), where \(\delta>0\) is the fractional part of …\(\pi\).  Then, in terms of \(\delta\):

\({\pi}^{3}=(3+\delta)^3=27+27 \delta+9 \delta^2+\delta^3\).

On the other hand…

\(3^{\pi}=3^{3+\delta}=27 \times 3^{\delta}=27 e^{\delta \log 3}\).

Now we make the observation that \(\log 3>1\), and use the expansion for the exponential function

\(e^x=1+x+\frac{1}{2} x^2+\frac{1}{6} x^3+\ldots\)

Given these facts, we can then state the following:

\(3^{\pi}>27+27 \delta+\frac{27}{2} \delta^2+\frac{9}{2} \delta^3>27+27 \delta+9 \delta^2+\delta^3\)

And therefore \(3^{\pi}>{\pi}^3\).