Is there a visual way to simplify $4\sqrt{12}+4\sqrt{27}$? I know the answer is $20\sqrt{3}$, but I want to geometrically explain it to a 14 year old.
Is there also a way to geometricaly interpret multiplication and division as well? Like $\sqrt{12}\cdot\sqrt{27}$ and $\frac{\sqrt{12}}{\sqrt{27}}$?
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Firstly, let's us not mix up things. Your equation reduces to $\sqrt{12}+\sqrt{27}=5\sqrt{3}$, which follows from $\sqrt{12}=2\sqrt{3}$ and $\sqrt{27}=3\sqrt{3}$. So how to see $\sqrt{12}=2\sqrt{3}$?
Draw a square with sidelength $\sqrt{3}$, so it's area is $3$. Take $4$ of them to form a square with side length is $2\sqrt{3}$. Now let your son compare: the side length of the big square is by construction $2\sqrt{3}$, on the other hand it's area is $4\cdot3$, so it's side length is $\sqrt{4\cdot3}$.
for multiplication, I would just construct a rectangle, which is easy since $\sqrt{n}$ is constructable.