Calculus 3 Optimization Pro question

This question is a little involved for a beginner, so please help if you believe you can do it. I am looking for the WAY to do it, not the answer please.

Question:

A company manufactures a product using x, y and z units of three different raw
materials. The quantity produced is given by the function
Q = 60x^(1/3) y^(1/4) z^(1/5) .
Suppose the cost of the materials per unit is $20, $15 and $24 respectively.

(a) Find the cheapest way to produce 6500 units of the product.

The answer is x=y=z=117.29,
however I am not exactly sure how to come to this conclusion. Please help me out, it is much appreciated. Thanks.

Using lagrange multipliers
60x^1/3*y^1/4*z^1/5
Cost = 20x+15y+24 z minimum with 60x^1/3y^1/4z^1/5-6500=0 (a)

so the function to minimize is
F(x,y,z,k) = 20x+15y+24z+k(60x^1/3y^1/4z^1/5-6500)
dF/dx= 20+20kx^-2/3=0
dF/dy=0 15+15ky^-3/4=0
dF/dz= 24+12kz^-4/5=0

and the equation (a) Solve for xyz eliminating k and going to (a)

santmann has the right method, but be careful on the partial differentials,

y and z terms are constants (when a differential with respect to x is being computed). They are being multiplied by a variable x, so you can't just drop them when you compute a differential with respect to x, for example, dQ/dx = 20x^-2/3 * (y^1/4 * z^1/5)