Irrational Exponents

Let's kick things up a notch. It's time to introduce irrational exponents. You'll never need another Red Bull again if you can get addicted to these bad boys. What does 2^π mean, for example? We can't possibly multiply π copies of 2 together, can we? Even if we could, wouldn't that get awfully messy?

While we can't write out all the digits of π, we can approximate π using decimals:

π ≈ 3 (not a great approximation)
π ≈ 3.1
π ≈ 3.14
π ≈ 3.141 and so on.

We can use these approximations of π to approximate 2^π.

2^π ≈ 2³
2^π ≈ 2^3.1
2^π ≈ 2^3.14
2^π ≈ 2^3.141

Since our approximations of π are rational numbers, we can find 2³, 2^3.1, and so on. We'll never be able to write down all the digits of 2^π, in the same way that we'll never be able to write down all the digits of π. Don't beat yourself up about it. Do what mathematicians do. Throw your hands in the air, give up on the thought of arriving at an exact answer, and PART-AY!

Uh, estimate. That last one was supposed to be "estimate."