Number of irrational roots

Question

I know that a number will have 2 square roots, 3 cube roots, and so on. This seems easy to extend to rational roots: e.g. There will be 3 2/3 roots.

But what about irrational roots?

How many roots are there to e.g. $3^{1/\pi}$? Is there a general solution?

What do you mean roots to $3^{1/\pi}$? There is no $x$ to solve for. — IraeVid, Jun 13 '23 at 13:59
@IraeVid Rephrased, how many complex numbers $z$ are there such that $z^\pi = 3$? More generally, given a value of $q$ and $p$ with $q$ an irrational number how many complex numbers $z$ are there such that $z^q = p$? Note that with if $q$ were rational written as $q=\frac{m}{n}$ in simplest terms, there are $n$ such values of $z$. — JMoravitz, Jun 13 '23 at 14:02
How do you define a rational root? i.e., how do you define a $\frac 23$ root? — 5xum, Jun 13 '23 at 14:03

score 3 · Accepted Answer · answered Jun 13 '23 at 14:08

There are always at most countably many values for such expressions, because they depend on the values of the logarithm. The possible values here are

$$z_k =e^{\frac1{\pi}(\ln 3 +2ki\pi)} = e^{\frac1{\pi}\ln 3}e^{2ki}=3^{1/\pi}\cos 2k + i3^{1/\pi}\sin 2k$$

Here, $\ln$ denotes the real-valued function of a positive real number.

The principal value is recovered with $k=0$, i.e., just the real number $3^{1/\pi}$.

1 Answers1