Prove that $5^{1/3}-3^{1/4}$ is irrational 
Prove that $5^{1/3}-3^{1/4}$ is irrational.

Let $x=5^{1/3}-3^{1/4}$. I started by:
\begin{align}
(x+3^{1/4})^3&=5\\
x^3+3^{3/4}+3x\cdot3^{1/4}5^{1/3}&=5\\
3^{3/4}+3x\cdot3^{1/4}5^{1/3}&=5-x^3 \tag{1}
\end{align}
Now I wished to prove by contradiction that - assuming $x$ is rational - LHS of (1) is irrational whereas RHS is rational, so the equation cannot be satisfied. However, the LHS is a sum of irrational terms, and we know that the sum of irrational terms is not always irrational.
I obviously don't wish to apply power four on (1) . Is there any other shorter method?

~An answer involving elementary field theory
If it were rational then we would have $5^{1/3} = q + 3^{1/4}$ where $q$ is a rational number.  This would then imply that the field $K = \mathbb{Q}(3^{1/4})$ would contain the field $\mathbb{Q}(5^{1/3})$ which obviously cannot be the case as  $[\mathbb{Q}(3^{1/4}):\mathbb{Q}] = 4$ and $[\mathbb{Q}(5^{1/3}):\mathbb{Q}] = 3$  and certainly $3$ does not divide $4$.
