<s>[INST] Prove that $(4/5)^{\frac{4}{5}}$ is irrational Prove that $(4/5)^{\frac{4}{5}}$ is irrational.
My proof so far:
Suppose for contradiction that $(4/5)^{\frac{4}{5}}$ is rational.
Then $(4/5)^{\frac{4}{5}}$=$\dfrac{p}{q}$, where $p$,$q$ are integers.
Then $\dfrac{4^4}{5^4}=\dfrac{p^5}{q^5}$
$\therefore$ $4^4q^5=5^4p^5$
I've got to this point and now I don't know where to go from here.
 [/INST] A slightly modified approach.
$(\frac{4}{5})^\frac{4}{5} = (\frac{4}{5})^{1-\frac{1}{5}} =  (\frac{4}{5})^1 \times(\frac{5}{4})^\frac{1}{5}$
Since the first factor is rational we need only show that the latter term $(\frac{5}{4})^\frac{1}{5}$ is irrational.
Let us assume that the latter is rational and write (where $p$ and $q$ are coprime)
$\frac{p}{q}=(\frac{5}{4})^\frac{1}{5} \implies 4\times p^5 = 5 \times q^5$
The L.H.S is even and this implies that $q$ is even (sub $q=2^5\times n$ in above); $q^5$ then must have a factor of $2^5$. This would then require $p$ to be even and thus violates the coprime assumption and gives the required contradiction.
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