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If n is an integer and x^n – x^-n = 0, what is the value of

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If n is an integer and x^n – x^-n = 0, what is the value of [#permalink] New post 05 Sep 2011, 11:17
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If n is an integer and x^n – x^-n = 0, what is the value of x ?

(1) x is an integer.
(2) n ≠ 0
[Reveal] Spoiler: OA

Last edited by Bunuel on 13 Jul 2013, 05:08, edited 1 time in total.
Renamed the topic and edited the question.
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Re: Value of X - Inequality!! [#permalink] New post 05 Sep 2011, 16:11
\(x^n - x^{-n}=0\)

\(x^n - \frac{1}{x^n}=0\)

\(x^n =\frac{1}{x^n}\)

\(x^{2n}=1\) (it can be assumed that \(x \neq 0\) because \(x^{-n}\) would then be undefined)

\(x^{2n}=1\) when:

1) \(x=1\)
2) \(n=0\) and \(x \neq 0\)

Statement 1) it cannot be determined whether \(x=1\), and no information is given on \(n\).

Yes example: \(x=1\), \(n=5\), and \(1^5 - 1^{-5} = 0\)

No example: \(x=5\), \(n=1\), and \(5^1 - 5^{-1} \neq 0\)

Statement 2) no information is given on \(x\). The same yes/no examples can be given to show that statement is insufficient.

Combined) \(n \neq 0\) and \(x\) is an integer, but we used the same example in both statements to demonstrate insufficiency.

[Reveal] Spoiler:
E
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Re: Value of X - Inequality!! [#permalink] New post 05 Sep 2011, 21:49
Expert's post
DeeptiM wrote:
If n is an integer and x^n – x^-n = 0, what is the value of x ?
(1) x is an integer.
(2) n ≠ 0


\(x^n = \frac{1}{x^n}\)

The equation a = 1/a is satisfied only when a = 1 or -1.

So, to satisfy this equation, \(x^n\) must be 1 or -1.

Statement 1:
If n = 0, \(x^n\) will be 1 for all integral values of x except 0. So x can take any value. We do not know the value of x. Not sufficient.

Statement 2:
If n ≠ 0, \(x^n\) will be 1 if x = 1 and \(x^n\) will be -1 if x = -1 and n is odd. So equation can be satisfied by both x = 1 and x = -1. We get two values for x. Not sufficient.

Using both statements together, we still have 2 values for x. Not sufficient.
Answer (E)
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Re: Value of X - Inequality!! [#permalink] New post 12 Sep 2011, 22:14
x^n - x^-n = 0 is possible only in either of the two ways
Case 1. n =0 (in this case value of x doesnt matter)
Case 2. x= -1 or +1 (in this case n can be any integer, since no power can change the total expression)

Now coming to the options (1) is insufficient from our deductions in case 2. x can be -1 or + 1

(2) is also insufficient since in case 1 and 2 we found that n can be of any integer


Together, (1) & (2) are insufficient because we still have x = -1 or +1 and n= any integer >0

Answer is E
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Re: Value of X - Inequality!!   [#permalink] 12 Sep 2011, 22:14
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