How do I multiply √3/2 * 1/√3 on paper?
Depends ...
If you mean: √(3)/2 x 1/√(3), then: [√(3) x 1 ] / [2 x √(3) ] = 1/2 (the √(3)s cancel).
If you mean: √(3/2) x 1/√(3), then: √(3/2) x √(1/3) = √(3/6) = √(1/2) = √(2)/2.