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?
