Review question

# If $2y = a^x + a^{-x}$, can we find $a^x$? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R6816

## Suggestion

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?