x^n = x^(n+2) for any integer n. Is it true that x > 0

[JbhB682]

Source : GMATPrep

OA : D

x^n = x^(n+2) for any integer n. Is it true that x > 0?

(1) x = x^2 - 2
(2) 2x < x^5

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Hi Sage -- Had a fundamental question on this problem

Regarding what's in the question stem specifically
-- x^n = x^(n+2) ----> this clearly tells me that x can either be [0,1 or -1] --> I get this

-- Then in the same question stem -- why is the GMAT then asking me if its true that x > 0 when it already has told me that x can be 0 | -1 or 1 ?

[Sage Pearce-Higgins]

Well, you can be sure that the GMAT is trying to confuse you. This is a good example of a problem with a greatly restricted number of possible cases. We're told (good rephrase) that x is -1, 0, or 1. Then the question asks whether x is positive. That's not ridiculous. Put the two pieces together and the rephrased question is simply "is x 1?".

why is the GMAT then asking me if its true that x > 0?

Remember, GMAT isn't asking you this. The problem that you need to solve is 'Do the pieces of data given in (1) and (2) provide enough information to give a certain answer to this question?'

[JbhB682]

Hi Sage

Reviewed some message boards and saw this is the fastest way

- Is this how you would solve it yourself ?

- If so, this is the first time i am seeing the process reversed
a) Normally we start with S1 and S2 and then apply it to the equations in the question stem

b) In this case however, you are doing the opposite -- and taking the information in the question stem first [x can be 0/-1 or 1] and plugging it into S1 and S2 to see if the equations in S1 and S2 even stand ...

Let me know your thoughts on this step by step

Is this how you would do it
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x^n = x^(n+2) for any integer n. Is it true that x > 0?

For x^n = x^(n+2) for ANY integer n, x must be 0, 1, or -1.

(S1) x = x^2 - 2 --> x^2 - x - 2 = 0 --> Only -1 satisfies x = x^2 - 2, thus x = -1 <0. Sufficient

(S2) 2x < x^5 --> from the possible values of x (0, 1, -1), only -1 satisfies 2x < x^5, thus x = -1 < 0. Sufficient.

Answer: D.

[Sage Pearce-Higgins]

- Is this how you would solve it yourself ?

Yes, this is the way I would solve it.

a) Normally we start with S1 and S2 and then apply it to the equations in the question stem

That depends. Here we have a question ("is x<0?"), but we also have some more data in the question stem ("x^n = x^(n+2)"). This is like a kind of 'extra statement' - it's a fact that we need to follow.

b) In this case however, you are doing the opposite -- and taking the information in the question stem first [x can be 0/-1 or 1] and plugging it into S1 and S2 to see if the equations in S1 and S2 even stand ...

No, this isn't accurate. I would describe the strategy used for this problem as "rephrasing the question". You're thinking 'what is this question really asking?' and narrowing things down before you get to applying the statements. Importantly, you're not 'seeing if S1 and S2 stand' - they are facts and we're seeing if the information they give is enough to answer the question. However, when we consider Statement 1, we also need to work within the constraints provided by the data at the beginning of the problem: we have to think of cases that agree with both Statement 1, and the fact that x^n = x^(n+2).
Look out for rephrasing / simplifying the question in some OG problems. For example, a question asks "Is xy > 0?". It's really asking "Do x and y have the same signs?".