Do we really know the value of expressions with irrational powers?

Question

The way we evaluate decimal powers such as $a^.75$ is by splitting it into $(a^3)$^(1/4). How then can we evaluate irrational powers? I know that we can approximate, but whenever we graph a^x we assume continuity at the irrational numbers, but how can we be certain of this?

Answer 1

The definition is that $a^b = e^{b \ln a}$ for positive $a$. This agrees on the rational points with what you get by roots and powers, and is continuous in $b$, so it is a decent definition.