NSolve will make a polynomial by adding auxiliary equations and variables to handle the radicals, e.g.
newvar[1][^100==x^133(so
newvar[1]is an alias for
x^1.33`. Suffice it to say that high degree polynomials lead to more work.
Here is an approximation that completes in 2-3 minutes.
Ee[X_] := 5.9 (X + 0.007)^(1/2 + 2/15) + 0.00413 X^(4/3) - 0.367;
NSolve[Ee[x] == 10, x, Reals]
(* Out[12]= {{x -> 2.68986312775}} *)
If we use 1/2+13/100
for that first exponent my guess is it will take considerably longer. I did not check to see whether specification of real solutions led to a significant change, either direction, in speed.