Freshman's dream

In mathematics, the freshman's dream, also known as freshman exponentiation, the child's binomial theorem, (rarely) the schoolboy binomial theorem, or the Frobenius identity is the generally-false equation (x + y)ⁿ = xⁿ + yⁿ. Beginning students commonly make this error in computing the power of a sum of real numbers, falsely assuming powers distribute over sums.

The correct result is given by the binomial theorem, which has additional terms in the middle when n ≥ 2. For example, when n = 2, the correct result is x² + 2xy + y², which can also be shown by multiplying (x + y)(x + y) by using the distributive property properly, or the FOIL method.

The freshman's dream is actually valid in commutative rings of characteristic p, such as the finite field $\mathbb {F} _{p}=\mathbb {Z} /p\mathbb {Z}$ , where p is a prime number, provided that the exponent n is p or more generally a power of p. Equivalently, the Frobenius map of the ring is an endomorphism. One way to prove this is to show that p divides all the binomial coefficients except for the first and the last, so all the intermediate terms are equal to zero. Another way to prove the common special case of this for $\mathbb {F} _{p}$ is to use Fermat's little theorem that a^p ≡ a mod p for all integers a. (This can be iterated for powers of p, using the property of exponentiation that taking a power of a power multiplies the exponents, and thereby proven in general using induction.)

The freshman's dream is valid for all n in tropical geometry (where multiplication is replaced with addition, so exponentiation becomes multiplication, and addition is replaced with minimum).

The freshman's dream equation is also true in some degenerate cases, such as when n = 1, when $n\geq 1$ and at least one of x and y is zero, and when n is an odd integer and $y=-x$ . These are all of the true cases for n ∈ {0, 1, 2, 3}, but when n ≥ 4 or n is negative or non-integer, there are generally additional pairs of complex numbers x, y that satisfy the equation.