You measure two quantities as A = 1.0 m ± 0.2 m, B = 2.0m± 0.2 m. We should report correct value for $\sqrt{\mathrm{AB}}$ as

Given, A = 1.0 m± 0.2 m, B = 2.0 m± 0.2m

Let $\mathrm{Y}=\sqrt{\mathrm{AB}}=\sqrt{(1.0)(2.0)}=1.414 \mathrm{m}$

Rounding off to two significant digit Y = 1.4 m

$\because \frac{\mathrm{\Delta Y}}{\mathrm{Y}}=\frac{1}{2}\left[\frac{\mathrm{\Delta A}}{\mathrm{A}}+\frac{\mathrm{\Delta B}}{\mathrm{B}}\right]=\frac{1}{2}\left[\frac{0.2}{1.0}+\frac{0.2}{2.0}\right]=\frac{0.6}{2\times 2.0}$

$\Rightarrow \mathrm{\Delta Y}=\frac{0.6\mathrm{Y}}{2\times 2.0}=\frac{0.6\times 1.4}{2\times 2.0}=0.212$

Rounding off to one significant digit, $∆\mathrm{Y} = 0.2 \mathrm{m}$
Thus, correct value for $\sqrt{\mathrm{AB}} = \mathrm{Y} + ∆\mathrm{Y} = 1.4 \mathrm{m} \pm 0.2 \mathrm{m}$