Need help simplifying complicated rational expression. Studying for my final and I can't figure this out.
Simplify:
$$\large\frac{\frac{3}{x^3y} + \frac{5}{xy^4}}{\frac{5}{x^3y} -\frac{4}{xy}}$$

The shortest route is to note that the least common multiple of all four denominators of the ‘small’ fractions is $x^3y^4$ and to multiply the expression by $1=\frac{x^3y^4}{x^3y^4}$:
$$\large{\frac{\frac{3}{x^3y} + \frac{5}{xy^4}}{\frac{5}{x^3y} -\frac{4}{xy}}\cdot\frac{x^3y^4}{x^3y^4}}\;.$$
I’ll let you check that this does the trick. It’s analogous to seeing that the least common multiple of $2,3,4$, and $6$ is $12$, so that multiplying
$$\large\frac{\frac12+\frac23}{\frac14+\frac56}$$
by $\frac{12}{12}$ will clean it up nicely.
