Given that \(2y = a^x + a^{-x}\), where \(a > 1\), \(x > 0\), prove that \[\begin{equation*} a^x = y + \sqrt{y^2-1}. \end{equation*}\]

What are we asked to find? Do we need to find \(x\) itself?

If, further, \(2z = a^{3x} + a^{-3x}\), prove that \[\begin{equation*} z = 4y^3 - 3y. \end{equation*}\]

We could just cube the expression for \(a^x\) and then work out \(a^{-x}\) and so on. But that seems quite painful. I wonder whether there is a simpler way?