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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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Bunuel
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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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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Hi Believer700.

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

believer700
Bunuel
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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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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Bunuel
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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shriramvelamuri
Hi Believer700.

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


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.
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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
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