Find the value of $\sqrt{5}$ correct to 2 decimal places; then use it to find the square root of ${\dfrac{3 - \sqrt{5}}{3 + \sqrt{5}}}$ correct to 2 significant digits.

Square root of 5 correct to two decimal places using division method:

$\sqrt{5} = 2.236 \approx 2.24$

${\dfrac{3 - \sqrt{5}}{3 + \sqrt{5}}}\\[1em] = \sqrt{\dfrac{(3 - \sqrt{5})(3 - \sqrt{5})}{(3 + \sqrt{5})(3 - \sqrt{5})}}\\[1em] = \sqrt{\dfrac{(3 - \sqrt{5})^2}{3^2 - (\sqrt{5})^2}}\\[1em] = \sqrt{\dfrac{(3 - \sqrt{5})^2}{9 - 5}}\\[1em] = \sqrt{\dfrac{(3 - \sqrt{5})^2}{4}}\\[1em] = {\dfrac{(3 - \sqrt{5})}{2}}\\[1em]$

Putting the value of $\sqrt{5}$,we get

$= {\dfrac{(3 - 2.24)}{2}}\\[1em] = {\dfrac{(0.76)}{2}}\\[1em] = 0.38$

Hence, $\sqrt{5} = 2.24$ and $\sqrt{\dfrac{3 - \sqrt{5}}{3 + \sqrt{5}}} = 0.38$.