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I am trying to find L$^*$ and K$^*$ (Labour and Capital respectively) given the following:

Q = $K^{1/4}$ $L^{3/4}$ where Q = 120, w=$24, and r=$128 (Not sure if you need w and r)

I know that if I can find either L$^*$ or K$^*$, I can find the other one easily.

I have found several questions relating to Cobbs-Douglas on here, but none where Q is given. A close example is this, but I don't have a budget constraint.

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    $\begingroup$ You can have tidy upper indices by putting whatever you want to have in the upper index in {} brackets. For example "K^{\frac{1}{4}}". $\endgroup$
    – Giskard
    Jun 2, 2015 at 18:28

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The problem is basically to derive the cost function of a firm. What you need to do is minimize costs given the output. That is $$ \min\limits_{K,L} w \cdot L + r \cdot K $$ subject to $$ K^{\frac{1}{4}} \cdot L^{\frac{3}{4}} = Q = 120. $$ You can solve this using Lagrange or the implicit function theorem (similar application as the MRS).

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