TITLE: What is $3^{\sqrt2}$? Definition of irrational Powers
QUESTION [6 upvotes]: What is $3^{\sqrt2}$ ?
Clearly we can say that $a^m = a \times a \times ...\times a \times a$ (m times) 
That is  $$3^2=3 \times 3 $$
But how can we define $3^{\sqrt2}$ ? How to understand the definition of irrational Powers ?
I do not need the value. I need a definition to understand this.

REPLY [6 votes]: I assume you're comfortable with exponentiation with rational exponents.
Let $(x_n)$ be a sequence of rational numbers such that $\lim x_n = \sqrt 2$.
Then $3^{\sqrt2} = \lim 3^{x_n}$.
One such sequence is
$$
x_{n+1} = \frac12\left(x_n + \frac2{x_n}\right), \quad x_0=2.
$$

REPLY [4 votes]: $3^{\sqrt{2}}= sup\{3^q \,| \, q \in \mathbb{Q} \wedge q^2<2 \}$