# Irrational^rational = rational

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

jgens
Gold Member
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:
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.

2 people
Mark44
Mentor
Is it true that in some cases an irrational number raised to a rational power is rational? If so what kinds of irrational numbers?
A very simple example is ##\pi^0 = 1##.
##\pi## is irrational, and 0 is rational.

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

HallsofIvy
Homework Helper
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.

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

arildno
Homework Helper
Gold Member
Dearly Missed
I think you just became a candidate for the Fields medal!

DrClaude
Mentor
I think you just became a candidate for the Fields medal!
No , I just became the candidate for another coffee!

Sorry Mark44, completely missed your humor there...

Mark44
Mentor
No , I just became the candidate for another coffee!

Sorry Mark44, completely missed your humor there...
That was arildno...

DrClaude
Mentor
That was arildno...
I thought you were trying to be funny by proposing the trivial case "to the power of 0".