Is there a closed form solution to $x^a + x^b - x^{a+b} = 0$ for $x > 0$, given $a,b > 0$?

Question

The title basically explains everything. Unfortunately I was stuck right from the beginning, so I cannot provide any more info. However, I provided a plot of the function for some values of $a$ and $b$.

Wolfram Alpha cannot find a solution for the general problem.

Answer 1

$$x^a+x^b-x^{a+b}=0$$

Let's investigate individual significant cases.

For $a,b\neq 0$, $x=0$ is a solution.

$x=1$ is not a solution.

For $a=0$ and for $b=0$, there is no solution.

For $a=b\neq 0$, $x=0$ and $x=2^{\frac{1}{a}}$ are solutions.

The equation is a transcendental equation for $x$ if $a$ or $b$ are not rational:

$$e^{\ln(x)a}+e^{\ln(x)b}-e^{\ln(x)(a+b)}=0.$$

If $a,b$ are algebraically independent, the equation depends on more than one algebraically independent monomials. If the inverse is an elementary function, we then cannot read the function expression of the inverse from the equation alone, according to the theorem in [Ritt 1925] which is also proved in [Risch 1979]. Besides, it's an open question when the general equation has solutions that are elementary numbers.

Lambert W doesn't seem to be applicable here for the general case.

If $b$ is a rational multiple of $a$, you get algebraic equations and you can use the known solution formulas for algebraic equations in some cases. [Ritt 1922] lists all kinds of algebraic equations that are solvable in radicals.
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[Risch 1979] Risch, R. H.: Algebraic Properties of the Elementary Functions of Analysis. Amer. J. Math. 101 (1979) (4) 743-759

[Ritt 1922] Ritt, J. F.: On algebraic functions which can be expressed in terms of radicals. Trans. Amer. Math. Soc. 24 (1922) (1) 21-30

[Ritt 1925] Ritt, J. F.: Elementary functions and their inverses. Trans. Amer. Math. Soc. 27 (1925) (1) 68-90

Answer 2

Apart from the obvious root $x=0$, there is no general closed-form solution for $x$. For instance, the nonzero root of $x+x^5-x^6=0$, which may be reduced to the quintic equation $x^5-x^4-1=0$, is famously not soluble with elementary functions. Solutions for this particular case are usually rendered with elliptic or hypergeometric functions, which are nonelementary.