Solving an equation with irrational exponents Is there any theory (analogous to Galois theory) for solving equations with irrational exponents like:
$ x^{\sqrt{2}}+x^{\sqrt{3}}=1$
?
 A: The study of such equations is not "abstract algebra" as it is usually understood.  The reason is that to even define the function $x^{\sqrt{2}}$, for example, requires analysis; one has to prove certain properties of $\mathbb{R}$ to ensure that such a function exists.  This is in marked contrast to the case of integer or rational powers, where one has a purely algebraic definition and the background theory is equational.  To define the function $x^{\sqrt{2}}$ one has to either define $e^x$ and the logarithm or consider a limit of functions $x^{p_n}$ where $p_n$ form a sequence of rational approximations to $\sqrt{2}$, and this is irreducibly non-algebraic stuff.
In particular, while polynomials can be studied in an absurdly general setting, transcendental equations like those you describe are more or less restricted to $\mathbb{R}$ (or $\mathbb{C}$ if you really want to pick a branch of the logarithm).  The LHS is an increasing function of $x$, so there is at most one root, which probably one can really only compute numerically if it exists.  (Its nonexistence can be ruled out by computing local minima in $(0, 1)$.)
This is another question which touches on a theme which has come up several times on math.SE, which is that exponentiation should really not be thought of as one operation.  Instead, it is a collection of related operations with various degrees of generality and applicability which happen to share the same algebraic properties, and one should not infer too much about how similar these operations are.  
A: 1.)
Your equations are algebraic equations in dependence of at least two algebraically independent monomials. With help of the main theorem in [Ritt 1925], that is also proved in [Risch 1979], we can conclude that the elementary function on the left-hand side of the equations doesn't have a partial inverse with non-discrete domain that is an elementary function. Therefore, the equations cannot be rearranged for $x$ by applying only elementary functions/operations we can read from the equation.
[Risch 1979] Risch, R. H.: Algebraic Properties of the Elementary Functions of Analysis. Amer. J. Math. 101 (1979) (4) 743-759
[Ritt 1925] Ritt, J. F.: Elementary functions and their inverses. Trans. Amer. Math. Soc. 27 (1925) (1) 68-90
2.)
Your equations are transcendental equations, namely exponential equations, namely exponential sum equations. There are some theorems for the solutions of exponential sum equations in some number fields.
3.)
Topological Galois Theory of Askold Khovanskii meets your question.
[Khovanskii 2014]:
"Vladimir Igorevich Arnold discovered that many classical questions in mathematics are unsolvable for topological reasons. In particular, he showed that a generic algebraic equation of degree 5 or higher is unsolvable by radicals precisely for topological reasons. Developing Arnold’s approach, I constructed in the early 1970s a one-dimensional version of topological Galois theory. According to this theory, the way the Riemann surface of an analytic function covers the plane of complex numbers can obstruct the representability of this function by explicit formulas. The strongest known results on the unexpressibility of functions by explicit formulas have been obtained in this way.
...
The monodromy group of an algebraic function is isomorphic to the Galois group of the associated extension of the field of rational functions. Therefore, the monodromy group is responsible for the representability of an algebraic function by radicals. However, not only algebraic functions have a monodromy group. It is defined for the logarithm, arctangent, and many other functions for which the Galois group does not make sense. It is thus natural to try using the monodromy group for these functions instead of the Galois group to prove that they do not belong to a certain Liouville class. This particular approach is implemented in one-dimensional topological Galois theory ..."
[Khovanskii 2014] Khovanskii, A.: Topological Galois Theory - Solvability and Unsolvability of Equations in Finite Terms. Springer 2014
[Khovanskii 2019] Khovanskii, A.: One dimensional topological Galois theory. 2019
[Khovanskii 2021] Topological Galois Theory - Slides 2021, University Toronto
Khovanskii's publications
4.)
Your equations are of the form
$$x^a+x^b=1\ \ \ (a,b\in\mathbb{C}).$$
This is a form similar to a trinomial equation. A closed-form solution can be obtained using confluent Fox-Wright Function $\ _1\Psi_1$ therefore.
Belkić, D.: All the trinomial roots, their powers and logarithms from the Lambert series, Bell polynomials and Fox–Wright function: illustration for genome multiplicity in survival of irradiated cells. J. Math. Chem. 57 (2019) 59-106
