<s>[INST] Write the expression so that the variable is only presented once and the exponents are positive. There is this problem 
$\frac{10x^{\frac{1}{3}}y^{-\frac{1}{4}}}{15x^{-\frac{1}{2}}y^{\frac{3}{4}}}$
and i know that the answer is 
$\frac{2x^{\frac{5}{6}}}{3y}$
but i cant find the logic of how is solved, Where do the coefficients come from?
I understand that fractions are the result of multiplication of expressions, but I do not understand how all come together. 
 [/INST] Note that $x^{r}x^{q} = x^{r+q}$, so if $r = -q$, then $x^{-q}x^{q} = x^{-q+q} = x^{0} = 1$. Thus, you can "remove" the negative exponent factors by multiplying the numerator & denominator by the factor using the corresponding positive exponent, as shown below
$$\begin{equation}\begin{aligned}
\frac{10x^{\frac{1}{3}}y^{-\frac{1}{4}}}{15x^{-\frac{1}{2}}y^{\frac{3}{4}}} & = \frac{2(5)x^{\frac{1}{3}}y^{-\frac{1}{4}}}{3(5)x^{-\frac{1}{2}}y^{\frac{3}{4}}} \left(\frac{x^{\frac{1}{2}}y^{\frac{1}{4}}}{x^{\frac{1}{2}}y^{\frac{1}{4}}}\right) \\
& = \frac{2x^{\frac{1}{3} + \frac{1}{2}}y^{\frac{-1}{4} + \frac{1}{4}}}{3x^{\frac{-1}{2} + \frac{1}{2}}y^{\frac{3}{4} + \frac{1}{4}}} \\
& = \frac{2x^{\frac{1}{3} + \frac{1}{2}}}{3y^{\frac{3}{4} + \frac{1}{4}}} \\
& = \frac{2x^{\frac{5}{6}}}{3y}\end{aligned}\end{equation}\tag{1}\label{eq1}$$
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