TITLE: Can you raise a number to an irrational exponent?
QUESTION [140 upvotes]: The way that I was taught it in 8th grade algebra, a number raised to a fractional exponent, i.e. $a^\frac x y$ is equivalent to the denominatorth root of the number raised to the numerator, i.e. $\sqrt[y]{a^x}$. So what happens when you raise a number to an irrational number? Obviously it is not so simple to break it down like above. Does an irrational exponent still have a well formed meaning?
The only example that comes to mind is Euler's identity, but that seems likes a pretty exceptional case. What about in general?

REPLY [2 votes]: The above answers are fine provided that you are only concerned with real numbers.  However, just as $x^{a/b}$ has $b$ solutions in the complex plane, $x$ to the power of an irrational will have an infinite number of solutions.