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

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Rounded to four decimal places, the square root of the square root of [#permalink]

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New post 27 Apr 2015, 04:31
4
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A
B
C
D
E

Difficulty:

  95% (hard)

Question Stats:

32% (01:35) correct 68% (01:31) wrong based on 241 sessions

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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


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Re: Rounded to four decimal places, the square root of the square root of [#permalink]

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New post 28 Apr 2015, 22:08
2
if \(x\) = \(\sqrt[4]{0.9984}\)
\(x^4\) = 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
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Rounded to four decimal places, the square root of the square root of [#permalink]

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New post 29 Apr 2015, 03:28
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4
Bunuel 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


Kudos for a correct solution.


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
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Re: Rounded to four decimal places, the square root of the square root of [#permalink]

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New post 29 Apr 2015, 04:27
1
Bunuel 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


Kudos for a correct solution.


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
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Re: Rounded to four decimal places, the square root of the square root of [#permalink]

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New post 29 Apr 2015, 10:33
Hi Believer700.

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

believer700 wrote:
Bunuel 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


Kudos for a correct solution.


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

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Re: Rounded to four decimal places, the square root of the square root of [#permalink]

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New post 04 May 2015, 04:59
Bunuel 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


Kudos for a correct solution.


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.
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Re: Rounded to four decimal places, the square root of the square root of [#permalink]

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New post 12 Mar 2016, 21:00
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<x<100
but all numbers in the answer choices fall in this range...

thus..I decided to take another approach..
start with the answer choices..square and see where the answer might be..
started with D:
9996 = 10,000-4
squared = 100000000 - 40,000 - 40,000 +16
100,000,000 -
40,000 -
40,000
99,920,016
so 0.9996 would be 0.992
so need smth smaller. E can eliminate right away

take C:
(10,000-6)(10,000-6) = 100,000,000 - 60,000 -60,000 +36
~0.9988 - this is smth similar. so picked C.
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Re: Rounded to four decimal places, the square root of the square root of [#permalink]

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New post 13 Mar 2016, 07:28
Bunuel 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


Kudos for a correct solution.

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.
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Re: Rounded to four decimal places, the square root of the square root of [#permalink]

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Re: Rounded to four decimal places, the square root of the square root of   [#permalink] 21 Apr 2018, 20:24
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