Is Square root your Confusion ?

Of course it's easy to do in the calculator, but let's try by hand. Note that the rational expression for $99.8001$ is $\frac{998001}{10000}$ The denominator is clearly $100^2$ whereas the numerator is $(1000-1)(1000-1) = 999^2.$ Taking the square root, we get $\sqrt{\frac{999^2}{100^2}} = \frac{999}{100} = 9.99$ Note the correct answer is actually $9.99$ which is closer to $10$ than $9.9$

$\sqrt{99.8001} = \sqrt{\dfrac{998001}{10000}} = \dfrac{999}{100} = \boxed{9.99}$

$9.99 \times 9.99 = 99.8001$

$999 \times 999= 1,000 \times 1,000 - 2(999) - 1^2 = 1,000,000 - 1,999 = \frac{998,001}{100^2} = 99.8001$

A pattern to notice:

$99 \times 99 = 9801$

$999 \times 999 = 998001$

$9,999 \times 9,999 = 99980001$

99.8001 is slightly less than 100, so its squareroot should be slightly less than 10. 10.1 is bigger than 10, and 8.9 and 9.1 are too low, their squares are nearer to 81. Thus, the answer is 9.9.

Do note that the real answer is 9.99, when you actually find the value instead of estimating.