You can use the first equation to isolate X in terms of Y. Substitute that X in to the second equation and then you can isolate Z in terms of Y. Substitute the X and Z in to the third equation to get an expression for Y. You can then work on just that expression in one variable.
When the floating point coefficients are converted to rational form, the expression for Y contains as a repeated sub-expression sqrt(56414126863825657569157648*Y^(1429/1000)-1614519984922625768635418*Y^(857/500)) . You can solve() this for Y and take the larger of the two solutions (the smaller is 0) and use that as an initial guess for your fsolve(). The actual solution is reasonably close by.
