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# For which of the following values of x is

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For which of the following values of x is [#permalink]

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24 Aug 2010, 11:41
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For which of the following values of x is $$\sqrt{1-\sqrt{2-\sqrt{x}}}$$ NOT defined as a real number?

A. 1
B. 2
C. 3
D. 4
E. 5
[Reveal] Spoiler: OA

Last edited by Bunuel on 16 May 2012, 02:53, edited 1 time in total.
Edited the question
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Re: NOT defined as a real number [#permalink]

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24 Aug 2010, 11:49
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Safiya wrote:
I'd be glad if someone could explain the logic for this question;

For which of the following values of x is

$$\sqrt{1-}$$$$\sqrt{2-}$$$$\sqrt{x}$$

NOT defined as a real number?

(A) 1
(B) 2
(C) 3
(D) 4
(E) 5

Real Numbers are: Integers, Fractions and Irrational Numbers. Non-real numbers are even roots (such as square roots) of negative numbers.

We have $$\sqrt{1-\sqrt{2-\sqrt{x}}}$$. For $${x=5}$$ expression becomes:$$\sqrt{1-\sqrt{2-\sqrt{5}}}$$ and $$2-\sqrt{5}<0$$, thus square root from this expression is not a real number.

Hope it helps.
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Re: NOT defined as a real number [#permalink]

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24 Aug 2010, 12:04
It helped indeed, thank you very much!
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Re: NOT defined as a real number [#permalink]

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20 Dec 2010, 00:18
Bunuel -

So the approach to problems such as this one is to work your way from the most inner root, out toward the main root? Always keeping track of whether the underlying roots are (1) negative (2) are larger than the main root.
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Re: NOT defined as a real number [#permalink]

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20 Dec 2010, 00:35
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tonebeeze wrote:
Bunuel -

So the approach to problems such as this one is to work your way from the most inner root, out toward the main root? Always keeping track of whether the underlying roots are (1) negative (2) are larger than the main root.

Consider another example: For which of the following values of x $$\sqrt{1-\sqrt{4-\sqrt{x}}}$$ is NOT defined as a real number?

A. 16
B. 12
C.10
D. 9
E. 4

First see whether $$4-\sqrt{x}$$ could be negative for some value of $$x$$ so you should test max value of $$x$$: $$4-\sqrt{x_{max}}=4-\sqrt{16}=0$$. As it's not negative then see whether $$1-\sqrt{4-\sqrt{x}}$$ can be negative for some value of $$x$$, so you should test min value of $$x$$ to maximize $$4-\sqrt{x}$$: $$1-\sqrt{4-\sqrt{x_{min}}}=1-\sqrt{4-\sqrt{4}}=1-1.41=-0.41<0$$.

Hope it's clear.
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Re: NOT defined as a real number [#permalink]

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20 Dec 2010, 01:02
Great visual explanation. I now understand. Thanks for taking the time to help.
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Re: NOT defined as a real number [#permalink]

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11 Jul 2011, 03:04
Bunuel wrote:
Safiya wrote:
I'd be glad if someone could explain the logic for this question;

For which of the following values of x is

$$\sqrt{1-}$$$$\sqrt{2-}$$$$\sqrt{x}$$

NOT defined as a real number?

(A) 1
(B) 2
(C) 3
(D) 4
(E) 5

Real Numbers are: Integers, Fractions and Irrational Numbers. Non-real numbers are even roots (such as square roots) of negative numbers.

We have $$\sqrt{1-\sqrt{2-\sqrt{x}}}$$. For $${x=5}$$ expression becomes:$$\sqrt{1-\sqrt{2-\sqrt{5}}}$$ and $$2-\sqrt{5}<0$$, thus square root from this expression is not a real number.

Hope it helps.

Why is 4 not the correct answer as it would yield root ( 2 - (root (4)) => root(2 - 2) = 0 .. Is root (0) not a real number?
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Re: NOT defined as a real number [#permalink]

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11 Jul 2011, 04:36
@siddhans, root(0) = 0, which is a real number, and the question is asking for a value that is not a real number. So 4 is not a correct choice.
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Re: For which of the following values of x is [#permalink]

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16 May 2012, 03:27
I don't understand how did you came from $$\sqrt{1-\sqrt{2-\sqrt{5}}} to 2-\sqrt{5}<0$$ ?
I would appreciate if you explain, because I'm obviously missing something.
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Re: For which of the following values of x is [#permalink]

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16 May 2012, 03:41
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Stiv wrote:
I don't understand how did you came from $$\sqrt{1-\sqrt{2-\sqrt{5}}} to 2-\sqrt{5}<0$$ ?
I would appreciate if you explain, because I'm obviously missing something.

If $${x=5}$$ then the expression becomes:$$\sqrt{1-\sqrt{2-\sqrt{5}}}$$. The expression under the second square root is $$2-\sqrt{5}$$. Now, since $$2-\sqrt{5}<0$$ then the square root from this expression is not a real number.

Hope it's clear.
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Re: For which of the following values of x is [#permalink]

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16 May 2012, 04:58
O yes,yes,yes... thanks! Sometimes I just look at numbers and don't see anything no matter how obvious it is.
I need a break!
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Re: For which of the following values of x is [#permalink]

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05 Jun 2013, 03:28
Bumping for review and further discussion*. Get a kudos point for an alternative solution!

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Re: For which of the following values of x is [#permalink]

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05 Jun 2013, 09:09
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Can we not simply take each value (calculated or not) under a radical, inside to out, and test if it's less than zero?

$$x<0$$ -- no answer choice satisfies

$$2-\sqrt{x}<0$$ -- otherwise reads $$2<\sqrt{x}$$ so $$x>2^2$$or 4....answer is E, no need to test [m]1-\sqrt{2-[square_root]x}[/square_root] as we have only one answer choice that meets this criteria. This approach is very similar to what Bunuel did but seems much quicker upfront in this case.
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Re: For which of the following values of x is [#permalink]

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11 Jul 2014, 21:19
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Re: For which of the following values of x is [#permalink]

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15 Jul 2015, 16:28
Hello from the GMAT Club BumpBot!

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Re: For which of the following values of x is [#permalink]

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15 Jul 2015, 20:29
Safiya wrote:
For which of the following values of x is $$\sqrt{1-\sqrt{2-\sqrt{x}}}$$ NOT defined as a real number?

A. 1
B. 2
C. 3
D. 4
E. 5

CONCEPT: Any Number is considered NON-REAL if the number under the square root is negative (i.e.√(-ve)) .

Such questions require us to check the smallest or highest values among option because at the extremes only the number will result in Real or NON-Real

Here, We need to check the highest value because x is being subtracted from other numbers

@x=5 (Option E)

√[1−√(2−√x)] = √[1−√(2−√5)] = √[1−√(2−2.2)] = √[1−√(-ve)] i.e. NON-REAL Number
Hence, Correct Option

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Re: For which of the following values of x is [#permalink]

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11 Jun 2016, 09:48
Nwsmith11 wrote:
Can we not simply take each value (calculated or not) under a radical, inside to out, and test if it's less than zero?

$$x<0$$ -- no answer choice satisfies

$$2-\sqrt{x}<0$$ -- otherwise reads $$2<\sqrt{x}$$ so $$x>2^2$$or 4....answer is E, no need to test [m]1-\sqrt{2-[square_root]x}[/square_root] as we have only one answer choice that meets this criteria. This approach is very similar to what Bunuel did but seems much quicker upfront in this case.

How would you solve this (the question mentioned by buneul above):
Consider another example: For which of the following values of x 1−4−x√‾‾‾‾‾‾‾√‾‾‾‾‾‾‾‾‾‾‾‾‾√1−4−x is NOT defined as a real number?

A. 16
B. 12
C.10
D. 9
E. 4
Re: For which of the following values of x is   [#permalink] 11 Jun 2016, 09:48
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