# Irrational^rational = rational

1. Feb 16, 2014

### caters

Is it true that in some cases an irrational number raised to a rational power is rational? If so what kinds of irrational numbers?

2. Feb 16, 2014

### jgens

Yes. For your first question note that √22 = 2. For your second question all number of this form should be algebraic.

Edit: As Mark44 pointed out below I need to add the caveat that the exponent be non-zero for my second claim to hold.

Last edited: Feb 16, 2014
3. Feb 16, 2014

### pwsnafu

It's also worth pointing out that irrational to the irrational can be rational.

The Gelfond-Schneider theorem says: if $a$ and $b$ are algebraic, with $a \neq 1,0$ and $b$ irrational, then $a^b$ is transcendental.

This means that $\sqrt{2}^{\sqrt{2}}$ is transcendental (hence irrational). Raise this to the power of $\sqrt{2}$ and you get 2.

4. Feb 16, 2014

### Staff: Mentor

A very simple example is $\pi^0 = 1$.
$\pi$ is irrational, and 0 is rational.

5. Mar 13, 2014

### rama

if the order of the root is say x, and the power you raise are multiples of x , then it becomes a rational number(or if the power is 0)
example:√2^4=4

6. Mar 13, 2014

### HallsofIvy

Staff Emeritus
Note that the irrational number has to be algebraic. A transcendental number raised to a rational power cannot be rational. In fact, it must still be transcendental.

7. Mar 13, 2014

### Staff: Mentor

8. Mar 13, 2014

### arildno

I think you just became a candidate for the Fields medal!

9. Mar 13, 2014

### Staff: Mentor

No , I just became the candidate for another coffee!

Sorry Mark44, completely missed your humor there...

10. Mar 13, 2014

### Staff: Mentor

That was arildno...

11. Mar 13, 2014

### Staff: Mentor

I thought you were trying to be funny by proposing the trivial case "to the power of 0".