I understand that exponents don't distribute over addition and have seen plenty of examples i.e. $$ (x + y)^2\neq x^2 + y^2 $$ but I'm wondering why that is. Multiplication distributes over addition e.g. $3(2+3) = 3(2) + 3(3)$ so if an exponent is just repeated multiplication why shouldn't the same be true for exponents?
i.e. why does $ (x + y)^2\neq x^2 + y^2 $
$\endgroup$ 52 Answers
$\begingroup$Multiplication distributes over addition [..], so if [exponentiation] is just repeated multiplication why shouldn't the same be true [..] ?
Multiplication distributes over addition, and if you repeatedly multiply by the same quantity or quantities, then that too distributes over addition: for example, $c(a+b) = ca + cb$ (doing multiplication once), $c^2(a+b) = c^2a + c^2b$ (multiplying by $c$ twice), $c^n(a + b) = c^n a + c^n b$ (multiplying $n$ times by by $c$).
But repeatedly multiplying $a$ with itself, repeatedly multiplying $b$ with itself, and finally adding the two is a different thing from repeatedly multiplying by $a + b$. There is no reason to expect that repeatedly multiplying by different things should somehow distribute over repeatedly multiplying by something else.
The quantity $(a + b)^n$ means that you repeatedly multiply it by $a + b$. The quantity $a^n$ means that you repeatedly multiply by $a$, and $b^n$ means that you repeatedly multiply by $b$. As you're multiplying by different things, the results can be arbitrarily different.
In the specific case of $(a + b)^2$ for instance, we can see that it is $(a + b) (a+b)$, and as a multiplication, it does distribute: $(a + b)(a + b) = (a+b)a + (a+b)b$. But $(a)(a) + (b)(b)$ is an entirely unrelated quantity, so it doesn't make sense to expect them to be equal.
$\endgroup$ $\begingroup$Perhaps this image answers your question:
