<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] You need to know some basic Exponential identities :
$$\frac{10x^{\frac{1}{3}}y^{-\frac{1}{4}}}{15x^{-\frac{1}{2}}y^{\frac{3}{4}}}$$
$$=\frac{10}{15} \times \frac{x^{1/3}}{x^{-1/2}} \times \frac{y^{-1/4}}{y^{3/4}}$$
Using $\frac{x^a}{x^b} = x^{a-b}$
$$=\frac{5 \times 2}{5 \times 3} \times x^{1/3+1/2} \times y^{-1/4-3/4}$$
$$=\frac{2}{3} \times x^{5/6} \times y^{-1}$$
$$=\frac{5 \times 2}{5 \times 3} \times x^{5/6} \times y^{-1}$$
$$=\frac{2x^{\frac{5}{6}}}{3y}$$
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