## 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 $$3^{\pi}>{\pi}^3$$.