freshman's dream error

The name ``freshman's dream theorem'' comes from the fact that people who are unfamiliar with mathematics commonly make the error of distributing exponents over addition and/or subtraction, typically when working in fields of characteristic zero. An example is the equation $(x+y)^2=x^2+y^2$ for $x,y \in \mathbb{R}$ . The equation is incorrect unless $x=0$ or $y=0$ . By no means does the exponent need to be a natural number or an integer for this error to occur. An example of this is the equation $\sqrt{x+y}=\sqrt{x}+\sqrt{y}$ for $x,y \in \mathbb{R}$ with $x \ge 0$ and $y \ge 0$ . This equation can be rewritten using the exponent $\frac{1}{2}$ , and again, the equation is incorrect unless $x=0$ or $y=0$ .

An easy way to explain to someone who is under the impression that exponents distribute over addition and/or subtraction is to provide a simple counterexample. For instance, when $x=3$ and $y=4$ , we have:

On the other hand, the freshman's dream theorem yields some instances in which exponents can be distributed over addition and/or subtraction.