find x

4.5333 = 1/x(1-(1/(1+x^10))) ....find x
Multiply both sides by x
subtract 1 from both sides
multiply both sides by (1+10^x)
add 1 to both sides
I think you end up with

4.5333X ¹¹ + 4.5333X - 10 ¹⁰ = 0

And if you can solve this you are doing really well.

You made a small mistake.
4.5333X ¹¹ + 4.5333X - X ¹⁰ = 0

From here out you can exclude x:
X.(4.5333X ¹⁰ + 4.5333 - X ⁹) = 0

So you can conclude that X=0 is one of the possibilities. However the equation is the beginning contained (1/X), so you would have to say that X=0 is not possible. Now you are left with:
4.5333X ¹⁰ + 4.5333 - X ⁹ = 0

Yes of course, thank you guest for picking that up.