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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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