You measure two quantities as A = 1.0 m ± 0.2 m, B = 2.0 m ± 0.2 m. We should report correct value for √(AB) as:

The correct option is 1.4m ± 0.2 m.

Let us determine the value of the quantity $X=\sqrt{AB}$ along with its uncertainty step-by-step.

Step 1: Calculate the mean value of $X$
Given:
$A=1.0\text{m}$
$B=2.0\text{m}$
Substituting these values into the formula:
$X=\sqrt{1.0\times 2.0}=\sqrt{2.0}\approx 1.414\text{m}$
Rounding to two significant figures (matching the precision of the measured quantities), we get:
$X\approx 1.4\text{m}$

Step 2: Calculate the fractional uncertainty in $X$
The expression for $X$ can be written as:
$X={\left(AB\right)}^{1/2}=A^{1/2}{B}^{1/2}$
Using the rules of error propagation for products and powers, the relative or fractional error in $X$ is given by:
$\frac{\Delta X}{X}=\frac{1}{2}(\frac{\Delta A}{A}+\frac{\Delta B}{B})$

Step 3: Substitute the known values to find the absolute uncertainty $\Delta X$
Here, the absolute uncertainties in $A$ and $B$ are:
$\Delta A=0.2\text{m}$
$\Delta B=0.2\text{m}$
Substituting these into the fractional error formula:
$\frac{\Delta X}{X}=\frac{1}{2}(\frac{0.2}{1.0}+\frac{0.2}{2.0})$
$\frac{\Delta X}{X}=\frac{1}{2}(0.2+0.1)=\frac{1}{2}\left(0.3\right)=0.15$

Now, calculate the absolute uncertainty $\Delta X$ by multiplying with the mean value of $X$:
$\Delta X=0.15\times X=0.15\times 1.414\text{m}\approx 0.212\text{m}$

Step 4: Report the final value with correct decimal places
Since the measured values of $A$ and $B$ are specified up to one decimal place, the uncertainty $\Delta X$ is rounded to one significant figure:
$\Delta X\approx 0.2\text{m}$
Therefore, the value of $\sqrt{AB}$ is reported as:
$1.4\text{m}\pm 0.2\text{m}$