# Thread: [SOLVED] Is this irrationnal ? Rationnal ? Or rationnaly interesting ?

1. ## [SOLVED] Is this irrationnal ? Rationnal ? Or rationnaly interesting ?

Hello !

Here is a little problem :

Proove that there exists two irrationnals a & b such as $\displaystyle a^b$ is a rationnal.

The proof is very easy, but you have to know it

2. I know what you are looking for. But I have a different solution. $\displaystyle e^{\ln 2} = 2$.

3. This is good too

Erm...is ln(2) irrationnal ?

I love the demo i read, because it's like...juggling with maths hihi

4. $\displaystyle \left( 2^\pi\right)^\frac{1}{\pi}$

5. Yes, this is the kind of demo it was given ^^

$\displaystyle ((\sqrt{2})^{\sqrt{2}})^{\sqrt{2}}$

6. Originally Posted by Moo
This is good too

Erm...is ln(2) irrationnal ?

I love the demo i read, because it's like...juggling with maths hihi
Suppose otherwise, then there exist integers $\displaystyle a$ and $\displaystyle b$ such that $\displaystyle e^{a/b}=2$.

Then:

$\displaystyle e=\root a \of{2^b}$

Now the left hand side is transcendental and the right hand side algebraic - a contradiction.

RonL