Solving an equation

Question

I need to solve the following equation (expr = 0) for "A", but looks like Solve and Reduce struggle. All parameters are positive, and "expr" is given below. (I suspect the issue might be due to powers not being simplified when "multiplied", and I am not sure how to best fix it). Many thanks in advance.

    expr = 1/2 (-((2 ((A - A R)^(-1/R))^(1 - R))/(-1 + R)) + (
   A (-r^2 (-1 + R) - (-1 + R) \[Mu]^2 + 
      2 R ((-1 + R) (A - A R)^(-1/R) - \[Rho]) \[Sigma]^2 + 
      2 r (-1 + R) (\[Mu] - R \[Sigma]^2)))/(R \[Sigma]^2))

Answer 1

You gave no assumptions on your parameters in your question. From the looks of things, you seem to be implicitly assuming that A is positive, and R is real:

FullSimplify[PowerExpand[expr], A > 0 && R ∈ Reals] /. (A - A R)^(-1/R) -> A^(-1/R) (1 - R)^(-1/R)
   A^(1 - 1/R) (1 - R)^(-1/R) R - A (r (-1 + R) + ρ) - (A (-1 + R) (r - μ)^2)/(2 R σ^2)

Solve[% == 0, A] // Simplify
   {{A -> -((2^R E^(I π R) R^R (-((r^2 (-1 + R) + (-1 + R) μ^2 + 2 R ρ σ^2 +
                                   2 r (-1 + R) (-μ + R σ^2))/(R σ^2)))^-R)/(-1 + R))}}