Rounded to four decimal places, the square root of the square root of

MANHATTAN GMAT OFFICIAL SOLUTION:

First, notice that the number you have to take the first square root of, 0.9984, is just a little less than 1, meaning that you could represent it as 1 – (something small).

Now, it’s hard to deal with square roots algebraically. But we can deal with their opposites – that is, squares. What would the square of 1 – (something small) be? Let’s write that as 1 – x, where we know that x is a small number, much less than 1.
(1-x)^2=1-2x+x^2

Now, since x is much less than 1, the x^2 term is much much less than 1. (To see why, imagine that x = 1/1,000. Then x^2 = 1/1,000,000.) Since we are rounding in this problem, we can make an approximation, dropping the x^2 term:

(1-x)^2=1-2x+x^2 ≈ 1 – 2x

Now we have the insight we need. Since the square of 1 – x is approximately 1 – 2x (doubling the gap between the number and 1) if x is very small, then we can go in the opposite direction: the square root of 1 – 2x is approximately 1 – x. In other words, you cut the gap between the number and 1 in half.

Write 0.9984 as 1 – 0.0016. In this case, 2x = 0.0016, so x = 0.0008.

The square root of 1 – 0.0016 is approximately 1 – 0.0008, or 0.9992.

Take the final step. The square root of 1 – 0.0008 is approximately 1 – 0.0004, or 0.9996.

You could also get to the answer by working backwards from the answer choices: the square of the square of the right answer must be approximately 0.9984. It will take longer, but brute force will get you there, eventually.

The correct answer is D.