# Irrational number to an irrational power

1. Jun 13, 2013

### Mabs

1. The problem statement, all variables and given/known data

if a and b are irrational numbers, is a^b necessarily an irrational number ? prove it.

3. The attempt at a solution
this is an question i got from my first maths(real analysis) class (college) , and have to say, i have only little knowledge about rational number, i would like to give it a try to find a solution but need some help, i know that a^b is not necessarily an irrational number cause if a = √2^√2 and b = √2 it becomes rational.

but i don't think its the proof that the question is expecting as its on real analysis. any help would be greatly appreciated and sorry for my English too.

2. Jun 13, 2013

### CAF123

You cannot prove that statement since, as you have shown via a counterexample, it is false. The question might have said:
'If a and b are irrational numbers, is ab necessarily irrational? Find a counterexample or prove it.​

3. Jun 13, 2013

### Mabs

sorry for that mate it actually says to prove my answer , yes or no. i know its no but i have no idea how to prove it.

4. Jun 13, 2013

### Staff: Mentor

Showing the counterexample you have proven the statement to be false. So the answer is "no" and the counterexample is the proof.

5. Jun 13, 2013

### collinsmark

Assuming that one also starts with the proof that (√2)√2 is irrational.

6. Jun 13, 2013

### Staff: Mentor

Good point

7. Jun 13, 2013

### epenguin

Don't need to prove that, since if it is rational we have found our counterexample. However if it is irrational, I don't see the proof that (√2)(√2)(√2) is rational.

8. Jun 13, 2013

### dextercioby

It's not the tower power of sqrt(2), rather

$$\left(\sqrt{2}^{\sqrt{2}}\right) ~ ^{\sqrt{2}}$$

which gives you the desired counterexample to the question in the problem, iff you manage to prove that the nr. inside the bracket is irrational.

Last edited: Jun 13, 2013
9. Jun 13, 2013

### collinsmark

Also a good point.

10. Jun 13, 2013

### epenguin

Independently of whether or not (√2)(√2)(√2) and ((√2)(√2))(√2) are the same thing.

If you did, how would it follow that your expression here is rational?

Last edited: Jun 13, 2013
11. Jun 14, 2013

### Mabs

thanks guys , proving sqrt2 ^sqrt2 is irrational , need to try on that one . thanks for the help.

12. Jun 14, 2013

### Mentallic

$$(\sqrt{2}^{\sqrt{2}})^{\sqrt{2}}=(\sqrt{2})^{\sqrt{2}\cdot \sqrt{2}}=(\sqrt{2})^2=2$$

13. Jun 14, 2013

### Curious3141

That's a toughie. Look up the Gelfond-Schneider theorem. With the theorem, it's trivial, but proving that theorem is very hard.

But as for the original question, this is a well-known proof that's often cited as a classic non-constructive proof. The proposition is that an irrational raised to an irrational power can be rational.

So we consider $x = \sqrt{2}^\sqrt{2}$. Either $x$ is rational or irrational. If it's the former, our work is done. If the latter, then let $y = x^\sqrt{2} = \sqrt{2}^2 = 2$. In this case, we've taken an irrational ($x$), raised it to an irrational power ($\sqrt{2}$) and the result is rational. QED.

It's a non-constructive proof because we didn't actually commit ourselves as to whether $x$ was rational or irrational. But we "covered our bases" either way.

14. Jun 15, 2013

### Mabs

thanks mate, can any one recommend me some books for additional reading on rational numbers ( from the basics ) , it would be great .