Denote $p=$ Price, $K=$ Number of robots, $L=$ Labour
$$\begin{cases}\dfrac{1}{3}pK^{-2/3}L^{1/3}= 2,\\ \\ \dfrac{1}{3}pK^{1/3}L^{-2/3}= 1.\end{cases}$$
Solve the above for $K$ and $L$.
The above two equations are the partial derivates of the product function:
$$f(K,L) = K^{1/3} L^{1/3}$$
It's a profit maximising exercise and I need to obtain the amount of inputs the firm hires. The answer is in the textbook but I just can't figure how they came up with it:
$$(K^{\ast},L^{\ast}) = (p^3/108 , p^3/54).$$
Thank you!
Cube both equations, getting ones that are in $K^2L$ and $KL^2$. Divide them to get one in $\frac KL$ and multiply them and cube root to get one in $KL$. Multiply those to get one in $K^2$.