We need to reduce each number to two significant figures:

- \(2.813 \times 10^{-3}\) becomes \(2.8 \times 10^{-3}\)

- \(1.063\) becomes \(1.1\)

- \(5.637 \times 10^{-2}\) becomes \(5.6 \times 10^{-2}\)

Now, we can substitute these values into the expression and simplify:

\[\frac{(2.8 \times 10^{-3} \times 1.1)}{(5.6 \times 10^{-2})} = \frac{3.08 \times 10^{-3}}{5.6 \times 10^{-2}} = \frac{3.08}{5.6} \times 10^{-1} = 0.55 \times 10^{-1}\]

Finally, we need to reduce the answer to two significant figures, which gives us 0.055.