## Manhattan GMAT Challenge Problem of the Week – 20 August 2012

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## Question

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

## Answer

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.

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

≈ 1 – 2*x*

Now we have the insight we need. Since the *square* of 1 – *x* is approximately 1 – 2*x* (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 – 2*x* 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, 2*x* = 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.**

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## 2 comments

Mahesh on August 26th, 2012 at 10:02 pm

The SqRt of 0.9984 is 0.99919 OR to approx 0.9992, someone missed (messed up)the steps.

prasoon on August 29th, 2012 at 4:03 am

No one has missed or messed up the steps.... IF u see correctly the sq. root of sq. root has been asked ... ie u need to calculate sq. root two times... that will come to 0.9996 as per the steps has explained...

the given explanation is totally fine...