# How do I solve this equation with Mathematica? [closed]

I've used Solve, Reduce, and NSolve, but I don't get anything

$$\frac{w^{3w-20}-w^{w}}{w^{w}-w}=w^{w}$$

any ideas?

• People here generally like users to post code as Mathematica code instead of just images or TeX, so they can copy-paste it. It makes it convenient for them and more likely you will get someone to help you. You may find this meta Q&A helpful Nov 23 '19 at 14:38

Either of these seem to do the trick:

eqn = (w^(3 w - 20) - w^w)/(w^w - w) == w^w;

Solve[eqn, w, Reals]//N
Reduce[eqn, w, Reals]//N
NSolve[eqn, w, Reals]


Given $$\frac{x^{3x - 20} - x^x}{x^x - x} = x^x$$ then dividing by $$x^x$$ leads to $$x^{2 x - 20} - x^x + x - 1 = 0.$$ Noticing $$x=-1$$ and $$x=1$$ are solutions then these should be expected results. For this version one could use the code

Solve[-1 + x - x^x + x^(-20 + 2 x) == 0 && x > -2, x, Reals]


It doesn't appear to be exactly 20 and 1 and -1 do not solve the original equation;

 myw = w /. NSolve[eqn, w, Reals, WorkingPrecision -> 100];
((w^(3 w - 20) - w^w)/(w^w - w) - w^w) /. w -> 20 // N
((w^(3 w - 20) - w^w)/(w^w - w) - w^w) /. w -> myw

Out[48]= 19.

Out[49]= {0.*10^-72}


Also, here's a complex solution:

myf[x_, n_] := Exp[x Log[Abs[x]] + I (Arg[x] + 2 n Pi)];
myf3[x_, n_] := Exp[3 x Log[Abs[x]] + I (Arg[x] + 2 n Pi)];
total[x_, n_] := (myf[x, n] x^(-20) - myf[x, n])/(myf[x, n] - x) -
myf[x, n];

myComplex = x /. FindRoot[total[x, 0] == 0, {x, 1 + I}]
total[myComplex, 0] // N

9.91725*10^-11 - 7.81348*10^-9 I