Every real number can be written as a **limit** of a sequence of rational numbers. For example, [itex]\pi[/itex] is the limit of the sequence 3, 3.1, 3.14, 3.1415, 3.14159, 3.141592, 3.1415926, etc. So the **definition** of, say, \(\displaystyle 2^{\pi}\) is the limit of the sequence \(\displaystyle 2^3\), \(\displaystyle 2^{3.1}\), \(\displaystyle 2^{3.14\), \(\displaystyle 2^{3.1415}\), etc. And each of those, of course, is 2 to a rational number power.