Freshman's dream

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The freshman's dream is a name sometimes given to the error (x + y)ⁿ = xⁿ + yⁿ, where n is a real number (usually a positive integer greater than 1). Beginning students commonly make this error in computing the exponential of a sum of real numbers.^[1]. When n = 2, it is easy to see why this is incorrect: (x + y)² can be correctly computed by FOILing as x² + 2xy + y². 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 case, the "mistake" actually gives the correct result, due to p dividing all the binomial coefficients save the first and the last. This theorem demonstrates that exponentiation by p produces an endomorphism, known as the Frobenius endomorphism of the ring.

[edit] Examples

$(1 + 4) 2 = 5 2 = 25$ , not $12 + 4 2 = 17$ . If we use the FOIL rule mentioned above, we obtain $17 + 8 = 25$ , which is correct.

$\sqrt{x^2+y^2}$ does not generally equal $\sqrt{x^2}+\sqrt{y^2}=x+y$ . For example, $\sqrt{9+16}=\sqrt{25}=5$ , which does not equal 3+4=7. In this example, the error is being committed with the exponent n = ½.