What integer is equal to the expression $\sqrt{50} - \sqrt{18} - \sqrt{8}$?

Question

I have no idea how to solve. Is there a formula or something?

Answer 1

$$\sqrt{50} - \sqrt{18} - \sqrt{8} = 5\sqrt{2} - 3\sqrt{2} - 2\sqrt{2} = 0.$$

Answer 2

Hint: $\sqrt{50} = 5\sqrt{2}$. Do the same for the other two radicals.

Answer 3

Since each term is irrational (being of the form $d\sqrt{m}$ where $m=2$ and $d$ is a positive integer), the only way that the sum can be rational is to be zero.

Proof:

If $a\sqrt{m}+b\sqrt{m}+c\sqrt{m} = n$, where $\sqrt{m}$ is irrational, and $n$ is an integer then $a\sqrt{m}+b\sqrt{m} = n-c\sqrt{m}$.

Squaring, $m(a+b)^2 =n^2-2nc\sqrt{m}+c^2m $ or $2nc\sqrt{m} =n^2+c^2m-m(a+b)^2 $ which is impossible since the left side is irrational (unless $n=0$) and the right side is an integer.

If $n=0$, then $c^2=(a+b)^2$ so that $c = -(a+b)$.