Taking √2 = 1.414, what is the value of 4/√2 correct to three decimal places?

Rationalise first rather than dividing by a decimal: 4/√2 = (4 × √2)/(√2 × √2) = 4√2/2 = 2√2. So the value is $2\times 1.414=2.828$. Multiplying by √2 instead of dividing gives $4\times 1.414=5.656$, and dividing 1.414 by 2 gives 0.707, which is 1/√2 — both come from mishandling the rationalising step.