How do we define power of irrational numbers?

Question

power of rational numbers for me can be defined as multiplying $m$ times the $nth$ root of $x$. because we have: $$ x^{\frac{m}{n}} $$ when : $m,n \in \Bbb Z $. Is this definition correct? if no what is the correct one and if yes, how can I extend this definition for irrational numbers? because we can't write them in the form $a/b$.

Answer 1

Assume $x^q$ is defined for $q \in\mathbb{Q}$. Given real numbers $x>0$, $y>0$ then $$x^y = sup \{x^q,q \in\mathbb{Q}, q<y \}$$ and extend it for $y<0$ for a full definition over $\mathbb{R}$.

Answer 2

I actually made a big post about extending exponentiation: see Generalization of the root of a number

Essentially, just as @charlus said - you define a sequence of rational numbers that converges to the irrational number you are interested in, and take a limit of the principal values. For example, if we had ${\pi}$, and a sequence of rational approximations of it ${\pi_n}$:

$${\pi_1 =\frac{3}{1},\pi_2 = \frac{31}{10}, \pi_3 = \frac{314}{100}...}$$

By construction it converges to ${\pi}$. Now, if we want to compute ${a^{\pi}}$:

$${a^\pi := \lim_{n\rightarrow\infty}a^{\pi_n}}$$

(note: you take the principal, positive real roots in the limit). You can also do this with any irrational quantity you want!