Why doesn't $(x+y)^2 $equal $x^2+y^2$?

Question

I need to explain this to someone.

I know obviously the expanded form gives you $x^2 + 2xy + y^2$ but technically don't the individual exponents multiply to give $x^2 + y^2$

I might say I'm looking for an interesting geometrical explanation for this.

Any help is appreciated!

Answer 1

I'm going to assume that you're familiar with the expansion of$$(a+b)(c+d)=a(c+d)+b(c+d)=ac+ad+bc+bd\tag1$$ With $(x+y)^2$, since the definition of an exponent is that $a^n=\underbrace{a\cdot a\cdot a\cdots a}_{n\text{ times}}$. Hence,$$(x+y)^2=(x+y)(x+y)=x(x+y)+y(x+y)=x^2+xy+yx+y^2\\=x^2+2xy+y^2$$

Answer 2

Source: https://letsplaymath.files.wordpress.com/2008/07/picture24.png

Does it explain your doubt?

Answer 3

Actually, it is true - sometimes. For fields of characteristic $2$ we have $(x+y)^2=x^2+y^2$. In general this has the nice name freshman'as dream, and this name indicates that we cannot expect it to be true in general (it would be a dream).

Answer 4

If $x=y=1$, then it should be obvious that

$$(1+1)^2=2^2=4$$

$$1^2+1^2=1+1=2$$

So it should be trivial that

$$(1+1)^2\ne1^2+1^2$$

And it cannot hold in general if it fails for at least one case.