# Mathematical definition of exponent [duplicate]

What does it mean by a^b in real number system? How is it defined mathematically?
It is clear in case of exponent being an integer.
i.e., a real number a is multiplied b times where b belongs to Z
If b is a rational..say b=p/q, then a^b can be interpreted as a^(1/q) multiplied p times.
But how is it defined when b is irrational?

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## marked as duplicate by T. Bongers, Daniel Fischer♦, apnorton, ncmathsadist, leoNov 30 '13 at 4:01

You use the density of rationals in $\Bbb R$.Find a sequence $x_n\to \sqrt 2$ and $x_n<\sqrt 2$ for every $n$. Same for a sequence $y_n\to \sqrt 2$ with $y_n>\sqrt 2$. Then ...?