Mental Math: Finding Square Roots to 1 Decimal Point

I'd just do good old Taylor series expansion up to linear term. $$ f(x+\Delta x) \approx f(x) + f'(x) \Delta x $$ so, in case of square roots $$ \sqrt{x + \Delta x} \approx \sqrt x + \frac {\Delta x}{2\sqrt x} $$ where $x$ - is the closest perfect square. Obviously, error might be huge if $\Delta x$ is big.

For example, $$ \sqrt{66} = \sqrt{64 + 2} \approx \sqrt{64} + \frac 2{2 \sqrt{64}} = 8 + \frac 18 = 8.125 $$ whereas $\sqrt{66} \approx 8.12403840463596 \ldots$

You should know all the perfect squares up to $100$, and the fact that $(n+\frac 12)^2=n(n+1)+\frac 14$. Then your best friend is $\sqrt{1+x}\approx 1+\frac x2$ for $x\ll 1$ Say you want the square root of $72$. You could say $72=8*9$, so the square root of $72.25$ is $8.5$ The odd quarter doesn't matter. If you want the square root of $68$, you have that $68=64(1+\frac 1{16})$, so $\sqrt{68}\approx 8(1+\frac 1{32})=8.25$ As the approximation is a little high, this rounds down to $8.2$, but it is close.