numerics - Any way of solving this system of nonlinear equations with non integer powers?

I have a system of four nonlinear equations. Some of the exponents are fractions. I was wondering if this is what is causing NSolve to run for hours without giving any results.

I first create a list of parameters that I eventually want to play around with to see how results change. This parameter list is "dat". I then specify the equations as follows:

dat = {alpha -> 1/3, beta -> 1/3, sigma -> 1/3, gam -> 0.5, 
  psy -> 0.5, delta -> 0.5, vu -> 0.5, a -> 1, A -> 1/3, B -> 1/3, 

  C -> 1/3, Ls -> 10, T -> 10, mc -> 1.5}

{p2 - w^psy pw^gam ((psy/gam)^gam + (gam/psy)^psy) == 0, 
  Ls - (beta/w)^(alpha + sigma) (pw/(alpha (1 + a C r T)))^
      alpha (r/sigma)^sigma (w Ls delta + r T A) - (pw psy/w gam)^
      gam ((w Ls vu + r T B)/p2) == 0, 
  T - (sigma/r)^(alpha + beta) (w/beta)^
      beta (pw/(alpha (1 + a C r T)))^alpha (w Ls delta + r T A) == 0, 
  pw - (((((1 + a C r T) alpha)/pw)^(beta + sigma) (w/beta)^
           beta (r/sigma)^

           sigma (w Ls delta + r T A) +  ((w gam)/(psy pw))^
           psy ((w Ls vu + r T B)/p2)) mc (1 + 
         a C r T))/((((1 + a C r T) alpha)/pw)^(beta + sigma) (w/
           beta)^beta (r/sigma)^
         sigma (w Ls delta + r T A)  + ((w gam)/(psy pw))^
         psy ((w Ls vu + r T B)/p2) (1 + a C r T)) == 0} /. dat

And then proceed with NSolve:

NSolve[%, {w, r, p2, pw}, Reals]

But Nothing happens. I tried using Rationalize but it does not seem to help. I am new to Mathematica so I hope that I am doing something wrong and that I missed something and can solve the system. Any suggestions?

Finally, how can I loop and redo this calculation with different parameters in the "dat" list?