Find the square root of 7 correct to two decimal places; then use it to find the value of $\sqrt{\dfrac{4 + \sqrt{7}}{4 - \sqrt{7}}}$ correct to three significant digits.

Square root of 7 correct to two decimal places using division method:

$\sqrt{7} = 2.645 \approx 2.65$

$\sqrt{\dfrac{4 + \sqrt{7}}{4 - \sqrt{7}}}\\[1em] = \sqrt{\dfrac{(4 + \sqrt{7})(4 + \sqrt{7})}{(4 - \sqrt{7})(4 + \sqrt{7})}}\\[1em] = \sqrt{\dfrac{(4 + \sqrt{7})^2}{4^2 - (\sqrt{7})^2}}\\[1em] = \sqrt{\dfrac{(4 + \sqrt{7})^2}{16 - 7}}\\[1em] = \sqrt{\dfrac{(4 + \sqrt{7})^2}{9}}\\[1em] = {\dfrac{(4 + \sqrt{7})}{3}}\\[1em]$

Putting the value of $\sqrt{7}$, we get

$= {\dfrac{(4 + 2.65)}{3}}\\[1em] = {\dfrac{(6.65)}{3}}\\[1em] = 2.22$

Hence, $\sqrt{7} = 2.65$ and $\sqrt{\dfrac{4 + \sqrt{7}}{4 - \sqrt{7}}} = 2.22$.