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

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Math Expert
Joined: 02 Sep 2009
Posts: 41913

Kudos [?]: 129537 [0], given: 12201

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

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27 Apr 2015, 04:31
Expert's post
11
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Difficulty:

95% (hard)

Question Stats:

31% (01:34) correct 69% (01:30) wrong based on 194 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

Kudos for a correct solution.
[Reveal] Spoiler: OA

_________________

Kudos [?]: 129537 [0], given: 12201

Manager
Joined: 15 May 2014
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Re: Rounded to four decimal places, the square root of the square root of [#permalink]

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28 Apr 2015, 22:08
1
KUDOS
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

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

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29 Apr 2015, 03:28
2
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2
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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:
$$1 - 0.9984 = 0.0016$$
$$0.0016 / 4 = 0.0004$$
$$1-0.0004 = 0.9996$$

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

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29 Apr 2015, 04:27
1
KUDOS
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

Kudos [?]: 25 [1], given: 210

Senior Manager
Joined: 27 Dec 2013
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Re: Rounded to four decimal places, the square root of the square root of [#permalink]

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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

_________________

Kudos to you, for helping me with some KUDOS.

Kudos [?]: 37 [0], given: 113

Math Expert
Joined: 02 Sep 2009
Posts: 41913

Kudos [?]: 129537 [1], given: 12201

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

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04 May 2015, 04:59
1
KUDOS
Expert's post
2
This post was
BOOKMARKED
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.

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

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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..
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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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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17 Aug 2017, 21:23
mvictor wrote:
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..
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.

Do such calculation intensive problems come in the GMAT?

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Re: Rounded to four decimal places, the square root of the square root of   [#permalink] 17 Aug 2017, 21:23
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