my question is, how did we get the exponent 7/12 for the K^7/12?

1. Suppose a firm is producing a product using the production function F(K, L) = 24L1/3K1/4, where each worker s paid $16 and each unit of capital costs $12. The firm has hired you to act as their consultant. Calculate the lowest total cost value ($) to produce 120 units of output in the LR using the Lagrange function. lowest total cost to produce 1zo units 2 = W. L Fr. K- > ( 24- 1 1/3 . Kin-q.) Mil W 2 2 = w- * ( 24 - 1/3 - 1-2/3. K /4) = 0 - W = 1 . 8. k'/ " - 1 = W. L 2/3 8. Klin 1 = = $ 1 24- 1/ 4 - K- 3/4. 1 1/3) = 0 - r= 1.6. 1 1/3 ->d=r.6 2/4 K3 ly 6. 1/3 12 = 24. 1 13 . 12/ = q. 1 1 = *2 - wol 7/3 rok bin 1 = 8rk 8. 12 . K - L=K 6 . Wul = frk -> 8. K'in 6 . 1 1/ 3 K = bul - K = L Er 2 4. 1 3 . K /4 = 9. = 120 24 . K 1 13 . K /4 = 120 ( 1 7 / 12 ) 97 = 15) 177 K = 15. 78 = 16 K = 1 = 16 TC = W . L + rok = 16. 16 + 12 . 16 = yug