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

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

Bunuel · Accepted Answer

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.

Harley1980 · Answer

Let's take middle variant and try calculate it's fourth square
Easy way to calculate this is make from  0.9994  number in form  (1-0.0006)  and square it twice
 (1-0.0006)*(1-0.0006) = 1 - 0.0006-0.0006 + ( something negligible, because of rounding ) = 1-0.0012 = 0.9988 
and square it one more time:
 (1-0.0012)*(1-0.0012) = 1 - 0.0012-0.0012 + ( something negligible, because of rounding ) = 1-0.0024=0.9976

And if we stop for a moment and look on this numbers we can quickly see pattern: 
fourth square of such numbers  0.9994  will be equal to  1 - 4 * (1-0.9994) = 1- 4*0.0006=1-0.0024=0.9976

We need number  0.9984  so let's reverse our pattern:
 1 - 0.9984 = 0.0016 
 0.0016 / 4 = 0.0004 
 1-0.0004 = 0.9996

And answer is D

sudh · Answer

if  x  =  \sqrt {0.9984} 
 x^4  = 0.9984

pick E and square the number twice
(0.9998)^2 
 = (10000- 2 )^2/10000^2
 = (100000000-2*2*10000+4)^2/10000^2; 4 is negligible
 = 9996/10000; the number 10000 in the numerator is reduced by  twice  the  2 
 = 0.9996
(0.9998)^4 
 = (0.9996)^2
 = (10000- 4 )/10000]^2
 = (100000000-2*4*10000+16)^2/10000^2; 16 is negligible
 = 9992/10000; the number 10000 in the numerator is reduced by  twice  the  4 
 = 0.9992

pick D and square the number twice
(0.9996)^2 
 = (10000-4)^2/10000^2
 = 0.9992
(0.9996)^4
 = (0.9992)^2
 = (10000-8)^2/10000^2
 = 0.9984

Answer D

believer700 · Answer

D..

Lets look it this way,

If we use Binomial

Then x = a -1/4(b)

where a=1
b= 1 - 0.9984
i.e b= 0.0016

Solving we get x = 0.9996

shriramvelamuri · Answer

Hi Believer700.

Please could you explain the procedure of bionomial. Sorry I am new to this concept and I would appreciate your help.

mvictor · Answer

i tried several methods..still..I believe it is way too tough... 0.9984 = 10000-16 which is a difference of 2 perfect squares. (100+4)(100-4) second is again a difference of 2 perfect squares: (100+4)(10-2)(10+2)/10000 now..we have 104x8x12/10000 find prime factorization: 4^4 * 3 * 13 / 10,000 if we take square root: 4^4 * sqrt(39) / 100 or 16*sqrt(39) / 100 now..sqrt 39 - is less than 49 - square of 7, but greater than 36 - square of 6. and it would be smth less than 6.5 suppose 6.1 => 6.2^2 = 38.44. not enough 6.3 -> 39.69 - too much so we are looking for a number between 16*6.2 and 16*6.3 but we can see that 16*16.25 = 100. then need to be smth smaller. 0.992

Mathivanan Palraj · Answer

The number is 9984/10000
We've to find 1/10 * square root of square root of 9984.
I used long division method, which I cannot present it here.
square root of 9984 is 99.92 approx
square root of 99.92 is 9.996 approx.
Therefore, the answer is 0.9996 approx.

Noida · Answer

Hi, We can use the Binomial Theorem to find the value of such expansions. We need to consider only the first term of the expansion and use that to find the approximate values. It states that: (1+x)^n = 1+nx (1-x)^n = 1-nx Say, we need to find which is greater: (1.01)^{10000} or 100 We can write this as (1+0.01)^{10000} = 1+(0.01*10000) = 101 Since 101 > 100, we know which value will be greater. Coming to the original question: 0.9984 = 1-0.0016 We have to take the fourth root of this: (1-0.0016)^{1/4} = 1-\frac{1}{4}*0.0016 = 1-0.0004 = 0.9996 Question: https://gmatclub.com/forum/of-the-follo ... 07334.html The attached image will help in understanding the expansion.

rahulbanka · Answer

Let the number be 1 – x (because the answers are in 0.999.. form, we can think of the number as 1 - something)

Since square roots are complex operations, we can solve the problem using reverse engineering—square rooting a number twice is the opposite of squaring another number twice)

Square of 1 – x ~ 1 – 2x (we can ignore x^2 since it is a small number)
Square of 1 – 2x ~ 1 – 4x (we can ignore 4x^2 since it is a small number)

Solve 1 – 4x = 0.9984, which will give x = 0.0004
Plug x in 1 – x to get D as the answer

Kinshook · Answer

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

(1-x)^n = 1 - nx + n(n-1)x^2/2
.9984^(1/4) = (1-.0016)^(1/4) = 1 -.0004 + (1/4)(-3/4)*(.0016)^2 = .9996 approximately

IMO D

ribbons · Answer

Another method that doesnt require such hefty math, check divisibility of 3, our original number is divisible by 3 so should its 4th root. Leaving us with only A and D. A ends in 0, which raised to the 4th power will also end with 0. D ends in 6, raised to 4 gives us 4 in units

Regor60 · Answer

6 TIMES 4 ends in 4

6 raised to any power>0 ends in 6

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

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