Is this a rational or irrational number?

Question

It is given that $$z=\sqrt\frac{\sqrt{3x+1}}{\sqrt{3x-1}}$$ How does one find whether $z$ is a rational or irrational number?

Answer 1

By sqaring twice we get:

$$ z^4 = \frac{3x+1}{3x-1} $$

To get a rational solution we are looking for two squared squares with a difference of two:

$$\begin{align} 3x+1 & = K^4 \\ 3x-1 & = L^4 \end{align}$$

With $L = K -1$ we get the smallest distance possible:

$$K^4 - (K-1)^4 = 2$$

This equation has no integer solution for $K$. Therefore, no rational value for $z$ can exist.

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Update:

As pointed out in the comments, $x = 17 / 45$ is in fact a solution.
So, what is wrong with my reasoning?