To find the coefficient of u^2 v^9 in the expansion of (2u^2 - 3v^3)^4, we can use the binomial theorem. The general term in the expansion of (2u^2 - 3v^3)^4 is given by:
T(r+1) = C(4,r) * (2u^2)^(4-r) * (-3v^3)^r
Here, r represents the index of the term, and C(4,r) represents the binomial coefficient (4 choose r) which is equal to 4!/(r!*(4-r)!).
We want to find the coefficient of u^2 v^9, so we need to find the term with r such that (4-r)*2 + 3r = 9. Solving for r, we get r=1.
Therefore, the term with r=1 is:
T(2) = C(4,1) * (2u^2)* (-3v^3)^3 = -216u^2 v^9
So the coefficient of u^2 v^9 in the expansion of (2u^2 - 3v^3)^4 is -216.
