The Intriguing Equation: αˣ + βˣ = γˣ

This expression is an exponential equation. Despite being a transcendental equation, its solution is very trivial for certain combinations of parameters. However, it is challenging to seek an analytical resolution. This is what we will present here using applying appropriate reasoning and techniques.

This is a variable transformation:

This will be driven into the equation by direct substitution

We can ungroup your terms

We know that the expression α² + β² = γ² (α > 0, β > 0, and γ > 0) can be associated with a three-sided polygon, and three very distinct cases can be attribute,

1. α + β = γ
2. α + β > γ
3. α + β < γ

Still in this sense, we can generalize these relations based on the law of cosines, that is:

and, depending on the angle between α and β will correspond to particularized situations.

1️⃣ ∠αβ = 180º:

or otherwise,

2️⃣ ∠αβ = 90º:

or otherwise,

3️⃣ α + β > γ and α = β:

4️⃣ α + β < γ: