TITLE: Real roots of $3^{x} + 4^{x} = 5^{x}$
QUESTION [5 upvotes]: How do I show that $3^{x}+4^{x} = 5^{x}$ has exactly one real root.

REPLY [3 votes]: Consider our equation as: 
$$\left(\frac{3}{5}\right)^x+\left(\frac{4}{5}\right)^x=1.$$
Notice that left side is a sum of 2 monotonically decreasing functions and their sum is a monotonically decreasing function. Hence, the only possible solution is x=2.
The proof is complete.