Talon Mcbride

2022-07-18

Simplifying ‘fraction-over-fraction-over-fraction’
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.
$x+\frac{1}{x+\frac{1}{x+\frac{1}{x}}}$

Steven Bates

Expert

$\begin{array}{rl}x+\frac{1}{x+\frac{1}{x+\frac{1}{x}}}& =x+\frac{1}{x+\frac{1}{\frac{{x}^{2}+1}{x}}}\\ & =x+\frac{1}{x+\frac{x}{{x}^{2}+1}}\\ & =x+\frac{1}{\frac{{x}^{3}+2x}{{x}^{2}+1}}\\ & =x+\frac{{x}^{2}+1}{{x}^{3}+2x}\\ & =\frac{{x}^{4}+3{x}^{2}+1}{{x}^{3}+2x}\end{array}$