Simplifying ‘fraction-over-fraction-over-fraction’

Question

It's repetitive and I'm just confused on what to do; I know we have to find a LCD but the fraction-over-fraction-over-fraction is throwing me off.

$$x + \frac{1}{x+\frac{1}{x+\frac{1}{x}}}$$

Answer 1

\begin{align} x+\frac{1}{x+\frac{1}{x+\frac{1}{x}}} &= x+\frac{1}{x+\frac{1}{\frac{x^2+1}{x}}}\\&= x+\frac{1}{x+\frac{x}{x^2+1}}\\&= x+\frac{1}{\frac{x^3+2x}{x^2+1}}\\&= x+\frac{x^2+1}{x^3+2x}\\&= \frac{x^4+3x^2+1}{x^3+2x} \end{align}