Finding Marshallian demands for Leontief production function with different powers
As Amit said, your answer is correct. As Toby said, you can rule out negative output. 

There is no much insight to add as an answer really. However, I was also fairly puzzled that such a simple problem gives a complex answer. As it turns out, you get a linear demand (on income) only if the exponents on both components is the same. 

Consider the more general case:

$$ U(x,y) = \text{min}\{ax^\alpha,by^\beta\} $$

Optimality requires (* notation for optimal values omitted)

$$ ax^\alpha = by^\beta\ $$

Replacing one of these in the budget constraint leads to:

$$ p_x x + p_y \left(\frac{a}{b}\right)^{\frac{1}{\beta}} x^{\frac{\alpha}{\beta}} = w $$

Here we see that only when $\alpha=\beta$, demand will be linear (on income). In this case:

$$ x = \frac{w}{p_x + p_y\left(\frac{a}{b}\right)^{\frac{1}{\alpha}}} $$

It seems an algebraic solution for the general (unrestricted) case cannot be derived, as the polynomial you get might have irrational powers.