What is the meaning of $2^\sqrt{3}$?

Question

What is the meaning of $2^\sqrt{3}$ ?

one can explain $2^3=2\times2\times2$

But how can I explain $2^\sqrt{3}$ etc. ?

How can I explain non-integer powers? I do not want the value. I need to know whether there is an explanation for this in real analysis.

Answer 1

One can also define $2^{\sqrt{3}}$ as follows. First, $2^k$ for integral $k$ is clear. If $k>0$ is an integer, it is clear how to define $2^{1/k}$: as the unique positive real $k^{\mathrm{th}}$ root of $2$. Then, for $r$ any positive real, choose a sequence $a_n = \frac{p_n}{q_n}$, where $p_n, q_n$ are integers, converging to $r$. Then $2^r:=\lim_{n\to\infty} 2^{a_n}$.

Answer 2

There are several ways to define $x^y$ for real $y$ and $x>0$. Here are three popular variations:

• $x^y=\exp(y\ln x)$, where $\exp$ is defined through, say, a power series
• $x^y=\lim_{n\to\infty}x^{y_n}$ where $y_n$ is a sequence of rationals converging to $y$, and $x^{y_n}$ is defined through roots and repeated multiplication
• The function $y\mapsto x^y$ is the unique continuous homomorphism from the group of real numbers with addition to the group of positive real numbers with multiplication which sends $1$ to $x$ (using the standard topology)

Either one gives the same result: $2^{\sqrt3}\approx 3.322$.

Answer 3

The meaning is $e^{\ln(2)\sqrt 3}=\exp(\ln(2)\sqrt 3)$. The exponential is a well defined function.