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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]

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Updated on: 13 Jul 2013, 06:08
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Difficulty:

75% (hard)

Question Stats:

52% (01:53) correct 48% (01:43) wrong based on 303 sessions

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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

Originally posted by DeeptiM on 05 Sep 2011, 12:17.
Last edited by Bunuel on 13 Jul 2013, 06:08, edited 1 time in total.
Renamed the topic and edited the question.
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Re: Value of X - Inequality!!  [#permalink]

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05 Sep 2011, 22:49
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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.
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Re: Value of X - Inequality!!  [#permalink]

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05 Sep 2011, 17:11
$$x^n - x^{-n}=0$$

$$x^n - \frac{1}{x^n}=0$$

$$x^n =\frac{1}{x^n}$$

$$x^{2n}=1$$ (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.

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Re: Value of X - Inequality!!  [#permalink]

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12 Sep 2011, 23:14
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

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Re: If n is an integer and x^n – x^-n = 0, what is the value of  [#permalink]

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06 Aug 2018, 04:13
x^n-1/x^n=0
x^2n-1/x^n =0
1. x is an integer
possible values
x=2 x=1 x=-1
n=0 n=1 n=1
Hence insufficient
2. From statement 2
still 2 possible values of x
x=1 x=-1
n=1 n=1
Hence insufficient
Combining, we will get same
x=1 x=-1
n=1 n=1
Hence insufficient
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Re: If n is an integer and x^n – x^-n = 0, what is the value of   [#permalink] 06 Aug 2018, 04:13
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