Problem Solving

allsaints921 wrote: Rounded to four decimal places, the square root of the square root of 0.9984 is approximately

A. 0.9990
B. 0.9992
C. 0.9994
D. 0.9996
E. 0.9998

While the algebraic ingredients that go into solving this question are certainly relevant on the GMAT, you won't see a question quite like this one on the test. Almost no one would know the math required in advance of seeing this type of problem, and to work out the mathematical basis needed to make an estimate here would take almost anyone well more than 2 minutes. But I can offer a different presentation of the solution in case it helps:

It's easier to first try the reverse problem, and a simpler version of it too. Say you are asked to approximate (0.97)^2. You could do long multiplication, but say instead we rewrite 0.97 as 1 - 0.03, so we want to find (1 - 0.03)^2. From algebra, we know that (x - y)^2 is always equal to x^2 - 2xy + y^2. Applying that here:

(1 - 0.03)^2 = (1)^2 - (2)(1)(0.03) + (0.03)^2
= 1 - (2)(0.03) + (0.03)^2

Now in this last line, if you think about how large each term is, the last term, (0.03)^2, is minuscule compared to the other two (it's equal to 0.0009, so it's tiny even compared to the middle term above). So if we only need to estimate, we can ignore that term, and we find that 0.97^2 is roughly equal to 1 - (2)(0.03) = 0.94.

From there you can generalize: if x is *very* close to zero, then (1-x)^2 will be very close in value to 1 - 2x. The larger x is, the worse this estimate will be. And if it's true that (1 - x)^2 is roughly equal to 1 - 2x, then taking square roots, 1 - x should be roughly equal to √(1 - 2x) when x is close to 0.

So that's what they've used in this question. It's easy to apply this estimate quickly, if you have numbers, if you notice that 1-x is just going to be halfway between 1 and 1-2x. Since we know 1-x is roughly equal to √(1 - 2x), that means √(1 - 2x) is roughly halfway between 1 and 1-2x. So if we need to estimate, say, √0.9, we can get a good estimate by finding the number which is halfway between 0.9 and 1, so 0.95 will be very close to the right answer (the actual answer is, to a few decimal places, 0.9487, so this is a very good estimate).

So to estimate the square root of 0.9984, we can just find its midpoint with 1; we get 0.9992. The question asks us to take the square root twice, so we can repeat the process, and the midpoint of 0.9992 and 1 is just 0.9996, so that will be a good approximation of the answer.

All of that said, the wording of the question is simply bizarre. The question asks "Rounded to four decimal places, the square root of the square root of 0.9984 is approximately". That question doesn't make sense. Rounded to four decimal places, the answer is exactly 0.9996. We're not approximating anything after we round off to four decimal places. As the question is worded, every answer is correct, because every answer is "approximately" equal to what we're asked to find. A question can ask you to round, or it can ask you to approximate, but it can't do both.