# Simplifying a Sum of Fractions by Removing Fractions in Denominators

I have a sum of fractions of the form $$x1 - \frac{(x2 + x3)}{(x4 + x5)} - \frac{(x6 + \frac{x7}{x13})}{(x8 - x9 - \frac{x10}{(x11 + x12)})}.$$

How might I simplify it to remove fractions from the denominator and then multiply the functions to get a single fraction (like Together would do) without simplifying the expression?

I would like something like this: $$\frac{(x1[x4 + x5][x13 (x8 - x9) (x11 + x12) - x10 (x13)] - [x2 + x3][ x13 (x8 - x9) (x11 + x12) - x10 (x13)] - [x6 (x13) + x7][ x4 + x5])}{[x4 + x5][x13 (x8 - x9) (x11 + x12) - x10 (x13)]}$$.

I was going to iterate over the terms with List and Denominator a number of times, find the denominators and multiply all the terms, but that quickly became unwieldy and I was unable to compactly multiply them.

Any suggestions or examples would be greatly appreciated!

exp = x1 - (x2 + x3)/(x4 + x5) - ((x6 + x7/x13)/(x8 - x9 - x10/(x11 + x12)))
Simplify[Together @ exp] // TeXForm


$$\small\frac{(\text{x11}+\text{x12}) (\text{x13} (\text{x4} (\text{x1} (\text{x9}-\text{x8})+\text{x6})+\text{x5} (-\text{x1} \text{x8}+\text{x1} \text{x9}+\text{x6})+(\text{x2}+\text{x3}) (\text{x8}-\text{x9}))+\text{x7} (\text{x4}+\text{x5}))-\text{x10} \text{x13} (-\text{x1} (\text{x4}+\text{x5})+\text{x2}+\text{x3})}{\text{x13} (\text{x4}+\text{x5}) (\text{x10}-(\text{x11}+\text{x12}) (\text{x8}-\text{x9}))}$$

Also

Apart[exp] // TeXForm


$$\small\frac{\text{x1} \text{x4}+\text{x1} \text{x5}-\text{x2}-\text{x3}}{\text{x4}+\text{x5}}+\frac{(\text{x11}+\text{x12}) (\text{x13} \text{x6}+\text{x7})}{\text{x13} (\text{x10}-\text{x11} \text{x8}+\text{x11} \text{x9}-\text{x12} \text{x8}+\text{x12} \text{x9})}$$

FullSimplify[Together @ exp] // TeXForm(* or *)
FullSimplify[Apart @ exp] // TeXForm


$$\small\text{x1}+\frac{(\text{x11}+\text{x12}) (\text{x13} \text{x6}+\text{x7})}{\text{x13} (\text{x10}-(\text{x11}+\text{x12}) (\text{x8}-\text{x9}))}-\frac{\text{x2}+\text{x3}}{\text{x4}+\text{x5}}$$

• Thank you very much for your speedy reply! Unfortunately, none of these methods do exactly what I hoped for. I would like to see which terms contribute to an expansion, so conserving grouping is important. If some of the terms are the same or differ only by coefficient, e.g. $x2+x3=x1(x4+x5)$ because $x4=x1, x5=x6, x2=x1^2, and x3=x1x6$, then Together and Apart would just combine them. Is there any way you know of to keep them separate? – BJParks Oct 9 '18 at 22:43