A firm has a production function, q=AL^aK^1-a, where 0<a<1. It wants to minimize cost for a given production q. The wage rate and rental rate on capital are w and r, respectively.
a. Write the Lagrangian expression for the cost minimization problem
b. Write the first-order conditions
c. Find the cost minimizing L and K

and

Consider a firm with the production function f(L,K) = L^0.5K^0.5. The wage rate and rental rate on capital are w and r, respectively.
a. Use the Lagrangian to derive the long-run cost function for this firm.
b. Suppose the government provides a subsidy of $10 per unit of capital to the fir. Rewrite the long-run cost function.

These are what I got for the answers

q1.

a) Larangian
min L = wL + rK + λ(q – ALaK1-a)


b) F.O.C
LL = w – λA(a)La-1K1-a = 0
LK = r – λA(1-a)LaK-a = 0
L λ = (q – ALaK1-a) = 0


c) w/r = λA(a)La-1K1-a / λA(1-a)LaK-a
w/r = aLa-1K1-a / 1-aLaK-a
w/r = aK / 1-aL
*sub 1-a for b*
K = (wbL/ar)

L λ = q = ALa(wbL/ar)b
(q = ALa(wbLb/arb)

I got stuck and dont know how to finish

q2.

a) Larangian:
L = wL0.5 + rK0.5 + λ[q – L0.5K0.5]

F.O.C:
LL = 0.5wL-0.5 – 0.5λL-0.5K0.5 = 0
LK = 0.5rK-0.5 – 0.5λK-0.5L0.5 = 0
L λ = q – L0.5K0.5 = 0

Combine first two conditions:
0.5wL-0.5 – λ0.5L-0.5K0.5 = 0.5rK-0.5 – λ0.5K-0.5L0.5
(wL-0.5 / rK-0.5) = (L-0.5K0.5 / K-0.5L0.5)
(wL-0.5 / rK-0.5) = (K / L)

Rearrange:
K = L(wL-0.5 / rK-0.5)
K = wL0.5 / rK-0.5

Substitute into third condition:
q = L0.5(wL0.5 / rK-0.5)0.5
q = (wL / rK-0.5)0.5
q = wL0.5 / rK-0.25

Solve for L:
L = (qrk-0.25/w)2
Solve for K:
K = (w2L/qr)-2

Cost function: C(w,r,q) = w(qrk-0.25/w)2 + r(w2L/qr)-2 = (qrk-0.25)2 + (w2L/q)-2




b) If the government grants a subsidy of $10 per unit of capital, the you replace all “K” with “k – 10”
[qr( k – 10)-0.25]2 + (w2L/q)-2

The question has too much detail

I don't understand? I basically solved it (I think) just looking to see if I went wrong anywhere