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 $x\geq 0$ and $y\geq 0$ . 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 counterexample. For instance, when $x=3$ and $y=4$ , we have:

$\begin{array}[]{ccccccc}(x+y)^{2}&=&(3+4)^{2}&=&7^{2}&=&49\\ \\ x^{2}+y^{2}&=&3^{2}+4^{2}&=&9+16&=&25\end{array}$

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

Title	freshman’s dream error
Canonical name	FreshmansDreamError
Date of creation	2013-03-22 16:07:23
Last modified on	2013-03-22 16:07:23
Owner	Wkbj79 (1863)
Last modified by	Wkbj79 (1863)
Numerical id	7
Author	Wkbj79 (1863)
Entry type	Example
Classification	msc 97D70