microeconomics!!! please helpppppp!!!?

Q = F(K,L) = 20 K^1/3 L^2/3
The price of capital is 8 per unit, and the price of labor is 4 per unit.

a) Find the cost-minimizing quantities of K and L if the fi…rm wants to produce ^Q units of output.
b) Derive the total cost function [Total cost of production as a function of Q]
c) Derive the marginal and average cost functions.
d) Compute the region of output over which economies of scale are present.[Hint: Economies of scale are present when AC decreases in Q]

a)
Q= 20 x ³√(KL²)
MPL=∂Q/∂L= (40/3) x ³√(K/L)
MPK=∂Q/∂K= (20/3) x ³√(L/K)²
Equilibrium: MPL/w = MPK/r
w=4
r=8
MPL/4 = MPK/8
(40/3) x ³√(K/L) / 4 = (20/3) x ³√(L/K)² / 8
(10/3) x ³√(K/L) = (10/3) x ³√(L/K)² / 4
³√(K/L) = ³√(L/K)² / 4
³√(L/K)² / ³√(K/L) = 4
L/K=4

L=4K
K=L/4

Q=F(K;L) → Q=F(L) → L=F(Q)
Q= 20 x ³√(KL²) = 20 x ³√(L³/4)
Q= 20L x ³√(1/4) = ³√2 x 10L
L = Q / (³√2 x 10) = Q / ³√2'000 ≈ 0.0793701Q ≈ Q/12.5992

Q=F(K;L) → Q=F(K) → K=F(Q)
Q = 20 x ³√(KL²) = 20 x ³√(K(4K)²)
Q = 20 x ³√(16K³) = 20K x ³√16
K = Q / (³√2 x 40) = Q / ³√128'000 ≈ 0.0198425Q ≈ Q/50.3968

Answ:
K = Q / ³√128'000
L = Q / ³√2'000

b)
TC(w;L;r;K) → TC(4;L;8;K) → TC(L;K) → TC(Q;K) → TC(Q)
TC = wL+rK
w=4
r=8
TC = 4L+8K
K = Q / ³√128'000
L = Q / ³√2'000
TC = (4Q / ³√2'000) + (8Q / ³√128'000) = 3Q/³√250

Answ: TC=3Q/³√250

c)
MC=∂TC/∂Q = 3/ ³√250 ≈ 0.47622
AC=TC/Q = 3/ ³√250 ≈ 0.47622
MC=AC

d)
There isn't economies of scale regions because AC function does not have downward-sloping ranges, TC function is linear and slopes upward over all range with constant slope coefficient ≈ 0.47622, thus AC and MC are horizontal lines and there aren't any economies-of-scale ranges.