Freshman's dream

The freshman's dream is a name sometimes given to the erroneous equation ${\displaystyle (x+y)^{n}=x^{n}+y^{n}}$, where ${\displaystyle n}$ is a real number (usually a positive integer greater than 1) and ${\displaystyle x,y}$ are non-zero real numbers. Beginning students commonly make this error in computing the power of a sum of real numbers, falsely assuming powers distribute over sums. When n = 2, it is easy to see why this is incorrect: (x + y)² can be correctly computed as x² + 2xy + y² using distributivity (commonly known by students in the USA as the FOIL method). For larger positive integer values of n, the correct result is given by the binomial theorem.

The name "freshman's dream" also sometimes refers to the theorem that says that for a prime number p, if x and y are members of a commutative ring of characteristic p, then (x + y)^p = x^p + y^p. In this more exotic type of arithmetic, the "mistake" actually gives the correct result, since p divides all the binomial coefficients apart from the first and the last, making all the intermediate terms equal to zero.

The identity is also actually true in the context of tropical geometry, where multiplication is replaced with addition, and addition is replaced with minimum.