# 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 FreshmansDreamError 2013-03-22 16:07:23 2013-03-22 16:07:23 Wkbj79 (1863) Wkbj79 (1863) 7 Wkbj79 (1863) Example msc 97D70