How do you compare $6-2\sqrt{3}$ and $3\sqrt{2}-2$? (no calculator)
Look simple but I have tried many ways and fail miserably.
Both are positive, so we cannot find which one is bigger than $0$ and the other smaller than $0$.
Taking the first minus the second in order to see the result positive or negative get me no where (perhaps I am too dense to see through).
Best Answer
$6-2\sqrt 3 \gtrless 3\sqrt 2-2$
Rearrange: $8 \gtrless 3\sqrt 2 + 2\sqrt 3$
Square: $64 \gtrless 30+12\sqrt 6$
Rearrange: $34 \gtrless 12\sqrt 6$
Square: $1156 \gtrless 864$