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Math Expert V
Joined: 02 Sep 2009
Posts: 56370

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14 00:00

Difficulty:   95% (hard)

Question Stats: 39% (01:53) correct 61% (02:07) wrong based on 152 sessions

### HideShow timer Statistics For a certain set of numbers, if $$x$$ is in the set, then both $$-x^2$$ and $$-x^3$$ are also in the set. If the number $$\frac{1}{2}$$ is in the set , which of the following must also be in the set?
I. $$-\frac{1}{64}$$

II. $$\frac{1}{64}$$

III. $$\frac{1}{\sqrt{2}}$$

A. $$I$$ only
B. $$II$$ only
C. $$III$$ only
D. $$I$$ and $$II$$ only
E. $$I$$, $$II$$ and $$III$$

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Math Expert V
Joined: 02 Sep 2009
Posts: 56370

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3
2
Official Solution:

For a certain set of numbers, if $$x$$ is in the set, then both $$-x^2$$ and $$-x^3$$ are also in the set. If the number $$\frac{1}{2}$$ is in the set , which of the following must also be in the set?
I. $$-\frac{1}{64}$$

II. $$\frac{1}{64}$$

III. $$\frac{1}{\sqrt{2}}$$

A. $$I$$ only
B. $$II$$ only
C. $$III$$ only
D. $$I$$ and $$II$$ only
E. $$I$$, $$II$$ and $$III$$

Since 1/2 is in the set, the so must be:

$$-x^2 = -\frac{1}{4}$$;

$$-x^3 = -\frac{1}{8}$$.

Since $$-\frac{1}{4}$$ is in the set, the so must be:

$$-x^3 =\frac{1}{64}$$

Since $$-\frac{1}{8}$$ is in the set, the so must be:

$$-x^2 =-\frac{1}{64}$$

The only number we cannot get is $$\frac{1}{\sqrt{2}}$$.

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What about - 1/2^(1/3)? Would it be in the set?
Math Expert V
Joined: 02 Sep 2009
Posts: 56370

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MaxOb12 wrote:
What about - 1/2^(1/3)? Would it be in the set?

No, we cannot say that -1/2^(1/3) must be in the set.
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Posts: 40

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Bunuel wrote:
Official Solution:

For a certain set of numbers, if $$x$$ is in the set, then both $$-x^2$$ and $$-x^3$$ are also in the set. If the number $$\frac{1}{2}$$ is in the set , which of the following must also be in the set?
I. $$-\frac{1}{64}$$

II. $$\frac{1}{64}$$

III. $$\frac{1}{\sqrt{2}}$$

A. $$I$$ only
B. $$II$$ only
C. $$III$$ only
D. $$I$$ and $$II$$ only
E. $$I$$, $$II$$ and $$III$$

Since 1/2 is in the set, the so must be:

$$-x^2 = -\frac{1}{4}$$;

$$-x^3 = -\frac{1}{8}$$.

Since $$-\frac{1}{4}$$ is in the set, the so must be:

$$-x^3 =\frac{1}{64}$$

Since $$-\frac{1}{8}$$ is in the set, the so must be:

$$-x^2 =-\frac{1}{64}$$

The only number we cannot get is $$\frac{1}{\sqrt{2}}$$.

I don't quite understand... the problem says if X is in the set, then -X^2 and -X^3 are in the set, but it doesn't say that -X^4 are also in the set ... which is what we are assuming to get from -1/4 to -1/64. Please help - thank you!
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Working towards 25 Kudos for the Gmatclub Exams - help meee I'm poooor
Math Expert V
Joined: 02 Sep 2009
Posts: 56370

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happyface101 wrote:
Bunuel wrote:
Official Solution:

For a certain set of numbers, if $$x$$ is in the set, then both $$-x^2$$ and $$-x^3$$ are also in the set. If the number $$\frac{1}{2}$$ is in the set , which of the following must also be in the set?
I. $$-\frac{1}{64}$$

II. $$\frac{1}{64}$$

III. $$\frac{1}{\sqrt{2}}$$

A. $$I$$ only
B. $$II$$ only
C. $$III$$ only
D. $$I$$ and $$II$$ only
E. $$I$$, $$II$$ and $$III$$

Since 1/2 is in the set, the so must be:

$$-x^2 = -\frac{1}{4}$$;

$$-x^3 = -\frac{1}{8}$$.

Since $$-\frac{1}{4}$$ is in the set, the so must be:

$$-x^3 =\frac{1}{64}$$

Since $$-\frac{1}{8}$$ is in the set, the so must be:

$$-x^2 =-\frac{1}{64}$$

The only number we cannot get is $$\frac{1}{\sqrt{2}}$$.

I don't quite understand... the problem says if X is in the set, then -X^2 and -X^3 are in the set, but it doesn't say that -X^4 are also in the set ... which is what we are assuming to get from -1/4 to -1/64. Please help - thank you!

The rule says that if a number is in the set, then minus that numbered squared is also in the set and minus that number cubed is also in the set.

So, if 1/2 is in the set, then so must be:

$$-x^2 = -\frac{1}{4}$$;

$$-x^3 = -\frac{1}{8}$$.

By the same logic, if $$-\frac{1}{4}$$ is in the set, then so must be:

$$-x^3 =\frac{1}{64}$$

By the same logic, if $$-\frac{1}{8}$$ is in the set, then so must be:

$$-x^2 =-\frac{1}{64}$$
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Manager  B
Joined: 18 Mar 2015
Posts: 120
Location: India
Schools: ISB '19
GMAT 1: 600 Q47 V26 GPA: 3.59

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i selected E for root 2 value must be around 0.0161 so it is in set where largest value is 1/2
Intern  B
Joined: 16 Jun 2018
Posts: 9

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Bunuel Why cant we go backwards.. like say if 1/2 exists in the set then (1/\sqrt{2}) and 1/[cube_root]2[/cube_root] also would be existing for 1/2 to exist in the set?
Pls explain
Math Expert V
Joined: 02 Sep 2009
Posts: 56370

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psych77 wrote:
Bunuel Why cant we go backwards.. like say if 1/2 exists in the set then (1/\sqrt{2}) and 1/[cube_root]2[/cube_root] also would be existing for 1/2 to exist in the set?
Pls explain

If x then y does not mean if y then x.
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Joined: 05 Oct 2018
Posts: 43
Location: United States
GMAT 1: 770 Q49 V47 GPA: 3.95
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1
Pm091989 wrote:
I think this is a high-quality question and I don't agree with the explanation. Since -1/8 exits, 1/64 exist but not -1/64.

-1/8 exists, so does -x^2, so we have -1*(-1/8)^2 = -1 * 1/64 = -1/64
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Joined: 29 May 2018
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As per me, a set by definition is a collection of finite numbers. If we go by the explanations, the given set can go on to become a collection of infinite numbers. Please correct me if my understanding is flawed.
Math Expert V
Joined: 02 Sep 2009
Posts: 56370

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Aastha2807 wrote:
As per me, a set by definition is a collection of finite numbers. If we go by the explanations, the given set can go on to become a collection of infinite numbers. Please correct me if my understanding is flawed.

Yes, this particular set will be infinite. The question asks which of the following MUST be in the set and both $$-\frac{1}{64}$$ and $$\frac{1}{64}$$ will definitely be in the set, though $$\frac{1}{\sqrt{2}}$$ will not necessarily be there.
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Joined: 10 Aug 2014
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bunuel,
pls explain how you got 1/64. i understand how you got -1/64.
thanks
Math Expert V
Joined: 02 Sep 2009
Posts: 56370

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Amureabc wrote:
bunuel,
pls explain how you got 1/64. i understand how you got -1/64.
thanks

Since 1/2 is in the set, then so must be $$-x^2 = -\frac{1}{4}$$;

Since $$-\frac{1}{4}$$ is in the set, then so must be $$-x^3 = -(-\frac{1}{4})^3=-(-\frac{1}{64})=\frac{1}{64}$$.
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