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Intern  Joined: 06 Aug 2007
Posts: 28
Location: Montreal
For which of the following values of x is (1 - (2 - x^(1/2))^(1/2))  [#permalink]

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50 00:00

Difficulty:   25% (medium)

Question Stats: 66% (00:57) correct 34% (01:03) wrong based on 1019 sessions

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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

Originally posted by Safiya on 24 Aug 2010, 12:41.
Last edited by Bunuel on 11 Oct 2019, 06:14, edited 2 times in total.
Edited the question
Math Expert V
Joined: 02 Sep 2009
Posts: 59144
For which of the following values of x is (1 - (2 - x^(1/2))^(1/2))  [#permalink]

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5
12
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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Intern  Joined: 06 Aug 2007
Posts: 28
Location: Montreal
Re: For which of the following values of x is (1 - (2 - x^(1/2))^(1/2))  [#permalink]

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It helped indeed, thank you very much!
Manager  Status: Current MBA Student
Joined: 19 Nov 2009
Posts: 90
Concentration: Finance, General Management
GMAT 1: 720 Q49 V40 Re: For which of the following values of x is (1 - (2 - x^(1/2))^(1/2))  [#permalink]

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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.
Math Expert V
Joined: 02 Sep 2009
Posts: 59144
Re: For which of the following values of x is (1 - (2 - x^(1/2))^(1/2))  [#permalink]

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11
5
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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GMAT 1: 720 Q49 V40 Re: For which of the following values of x is (1 - (2 - x^(1/2))^(1/2))  [#permalink]

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Great visual explanation. I now understand. Thanks for taking the time to help.
Manager  Joined: 16 Feb 2012
Posts: 144
Concentration: Finance, Economics
Re: For which of the following values of x is (1 - (2 - x^(1/2))^(1/2))  [#permalink]

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1
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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Math Expert V
Joined: 02 Sep 2009
Posts: 59144
Re: For which of the following values of x is (1 - (2 - x^(1/2))^(1/2))  [#permalink]

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1
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 (1 - (2 - x^(1/2))^(1/2))  [#permalink]

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O yes,yes,yes... thanks! Sometimes I just look at numbers and don't see anything no matter how obvious it is.
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Re: For which of the following values of x is (1 - (2 - x^(1/2))^(1/2))  [#permalink]

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1
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 (1 - (2 - x^(1/2))^(1/2))  [#permalink]

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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

We need to find the value of x that leaves us with a negative value under the root sign, which would not be defined as a number.

We see that if x = 5, then 2 - √5 < 0, so √(2 - √5) will not be a real number.

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_________________ Re: Sqrt Q   [#permalink] 11 Oct 2019, 06:11
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