When to use these fractions properly? I'm self-taught in mathematics, however, I have missed many basic ideas of arithmetic, where I'll usually use symbolab to help me get through some simplification of equations. although, I'm starting to get annoyed by this, as its impeding my learning. I would really like your support in helping me understand the distinction in fractions likeso:
$$[a] \space \cfrac{x}{1-x\over x}$$ when this could be simplified as either:
$$[1] \space\frac{x}{(1-x)x} $$
or $$[2] \space \frac{x^2}{1-x}$$
Though, I've also used symbolab and it's given me fractions likeso:
$$[b] \space \cfrac{x \over1+x}{1-x\over x}$$
Where it will be split like the first fraction into either (1) or (2). Are there rules that help me decide if its either (1) or (2) given equation (a), and when is it acceptable to use these for equation (b).
Or does it not matter for (b), and that the result should ensue likeso:
$$[b] \space \cfrac{x \over1+x}{1-x\over x} = \frac{x}{1+x}\frac{1-x}{x}$$
Thank you for the support! this should help me from relying on symbolab all the time for this.
$$\frac{x}{\frac{y}{z}}=\frac{(x)}{\left(\frac{y}{z}\right)}=x\left/\frac{y}{z}\right.=\frac{xz}{y}$$
$$\frac{\frac{a}{b}}{\frac{c}{d}}=\frac{\left(\frac{a}{b}\right)}{\left(\frac{c}{d}\right)}=\frac{a}{b}\left/\frac{c}{d}\right.=\frac{ad}{bc}$$
In both cases, the middle fraction line is the &quot;longest&quot;, so there are implicit brackets around the numerators and the denominators.
2 is right I think. This is just a pure notation question. The situation tells for sure. Use common sense lol.