Suppose that a firm produces a good using capital, skilled labor, and unskilled labor. Let $K$ denote the amount of capital,$L_1$ unskilled labor, $L_2$ skilled labor. The production function is $f(L_1, L_2, K) = K^2 min\{L_1, L_2^{\frac{1}{3}}\}$. Furthermore, let capital rental rate, $r=200$, unskilled wage rate is $w_1=5$, and skilled wage rate is $w_2=6$.
Find long run cost function.
I am unsure as to how to approach this problem given the K^2 term in the function. Thus, I tried when $K=1$. However, I would like to know how to approach it for all $K$, where $K$ is a natural number.
My Attempt
Assume $K=1$. Then we have $$f(L_1, L_2, 1) = min\{L_1, L_2^{\frac{1}{3}}\}$$
From here we can see that in order to minimize cost at a particular level $q$, we have $L_1=L_2^{\frac{1}{3}}=q$ which leads to $$L_1=q$$ $$L_2=q^3$$. Thus, $$c(q)=200+5q+6q^3$$