What does $2^x$ really mean when $x$ is not an integer? We all know that $2^5$ means $2\times 2\times 2\times 2\times 2 = 32$, but what does $2^\pi$ mean? How is it possible to calculate that without using a calculator? I am really curious about this, so please let me know what you think.

Euler's identity would be another use of exponents outside of integers as there are some complex numbers used in the identity.  Euler's formula explains how to evaluate such values which does help in some cases to evaluate the function.
