Which is bigger, $1+3\sqrt{2}$ or $3\sqrt{3}$? Which is bigger, $a = 1+3\sqrt2$ or $b = 3\sqrt3$?
To find out result, I am doing: $(3\sqrt3)^2-(1+3\sqrt2)^2=8-2\sqrt{18}$.
But how can I find if the value of $8-2\sqrt{18}$ is positive or negative?
Thank you.

You can use an “unknown relation”: denote by ? either $<$ or $>$; so long as you do operations that don't change the direction of inequalities, the symbol will always mean the same; so, when you're done you'll know which one it is.
\begin{align}
1+3\sqrt{2}&\mathrel{?}3\sqrt{3}\\
(1+3\sqrt{2})^2&\mathrel{?}(3\sqrt{3})^2\\
1+6\sqrt{2}+18&\mathrel{?}27\\
6\sqrt{2}&\mathrel{?}27-1-18\\
6\sqrt{2}&\mathrel{?}8\\
3\sqrt{2}&\mathrel{?}4\\
(3\sqrt{2})^2&\mathrel{?}16\\
18&\mathrel{?}16
\end{align}
So ? was $>$.
