Need help simplifying complicated rational expression. Studying for my final and I can't figure this out.
Simplify:
$$\large\frac{\frac{3}{x^3y} + \frac{5}{xy^4}}{\frac{5}{x^3y} -\frac{4}{xy}}$$
 A: 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.
A: Find the LCM of all four denominators.  It is just the product of the highest power of $x$ and the highest power of $y$ that appear.  Now multiply numerator and denominator by that LCM.  It will clear all the fractions except the main one.
A: Hint: Is there something that you could multiply both the numerator and denominator by in order to make this expression easier to tackle? Then it might be easier to simplify and reduce the expression.
(Bigger hint: multiply by $\frac{x^{a}y^{b}}{x^{a}y^{b}}$ for some appropriate choice of $a, b$ and then simplify!)
A: $$\large\frac{\frac{3}{x^3y} + \frac{5}{xy^4}}{\frac{5}{x^3y} -\frac{4}{xy}}$$
$$\large\frac{\frac{3y^3+5x^2}{x^3y^4}}{\frac{5-4x^2}{x^3y}}$$
$$\dfrac{3y^3+5x^2}{x^3y^4}\cdot{\dfrac{x^3y}{5-4x^2}}$$
$$\dfrac{3y^3+5x^2}{{y^3}\cdot{(5-4x^2)}}$$
