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

Question

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

KarishmaB · Accepted Answer

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)

bodleyev · Answer

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 
eq 0  because  x^{-n}  would then be undefined)

x^{2n}=1  when:

1)  x=1 
2)  n=0  and  x 
eq 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} 
eq 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 
eq 0  and  x  is an integer, but we used the same example in both statements to demonstrate insufficiency.

E

raghavakumar85 · Answer

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

Raksat · Answer

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
Answer E

Kinshook · Answer

If n is an integer and x^n – x^-n = 0, what is the value of x ?
x^n – x^-n = 0 
x^{2n} = 1
If n=0; Any x will satisfy x^{2n} = 1
If n≠ 0; x = 1 or -1

(1) x is an integer.
If n=0; Any x will satisfy x^{2n} = 1
If n≠ 0; x = 1 or -1
NOT SUFFICIENT

(2) n ≠ 0
x = 1 or -1
NOT SUFFICIENT

(1) + (2) 
(1) x is an integer.
(2) n ≠ 0
x = 1 or -1
NOT SUFFICIENT

IMO E

Basshead · Answer

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

x^{n} – \frac{1}{x^{n}} = 0

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

This is true when:
1)  x^{n}  = -1, 1
2)  n = 0

Lets take a look at our statements:

(1) x is an integer.

x can be -1 or 1; x can be any value if n = 0. Insufficient.

(2) n ≠ 0

if n ≠ 0, x can still be -1 or 1.

(1+2) x is an integer AND n ≠ 0

x can be -1 or 1. Insufficient.

Answer is E .

BrentGMATPrepNow · Answer

Given : n is an integer and  x^n – x^{-n} = 0   
In other words:  x^n = x^{-n}

----------ASIDE----------------
Important: Many students will incorrectly conclude that, since  x^n = x^{-n} , it must be the case that  n = -n  . However, this is true only if  x 
eq 0 ,  x 
eq 1  and  x 
eq -1 
To illustrate this point, consider the equation  1^2 = 1^5 
Even though the two powers have the same base (1), we certainly can't conclude that  2 = 5

Likewise, if  0^m = 0^n , we can't then conclude that  m=n 
-----------------------------

So, if  x^n = x^{-n} , then there are 4 possible cases:
  case a :  x 
eq 0 ,  x 
eq 1  and  x 
eq -1 , and  n = -n   (which means  n = 0 ) 
 case b :  x = 0 , and  n  can have any positive value 
 case c :  x = 1 , and  n  can have any value 
 case d :  x = -1 , and  n  is an even integer

Target question :    What is the value of x?

Statement 1 : x is an integer  
There are several values of x (and n) that satisfy statement 1. Here are two: 
 Case a : x = 1 and n = 1. In this case, the answer to the target question is  x = 1 
 Case b : x = -1 and n = 2. In this case, the answer to the target question is  x = -1 
Since we can’t answer the  target question  with certainty, statement 1 is NOT SUFFICIENT

Statement 2 : n ≠ 0 
PRO TIP: Before I start testing various values of x and n, I first check to see whether I can reuse any of the values I tested for statement 1....it turns out that both pairs of values I used in statement 1 also satisfy statement 2. That is....
 Case a : x = 1 and n = 1. In this case, the answer to the target question is  x = 1 
 Case b : x = -1 and n = 2. In this case, the answer to the target question is  x = -1 
Since we can’t answer the  target question  with certainty, statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined    
Since I was able to use the  same counter-examples  to show that each statement ALONE is not sufficient, the same counter-examples will satisfy the two statements COMBINED. 
In other words, 
 Case a : x = 1 and n = 1. In this case, the answer to the target question is  x = 1 
 Case b : x = -1 and n = 2. In this case, the answer to the target question is  x = -1 
Since we still can’t answer the  target question  with certainty, the combined statements are NOT SUFFICIENT

Answer: E

Cheers,
Brent

TheBipedalHorse · Answer

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 – x^(-n) = 0
no matter what value of n we use, the entire equation will always be in the form of (x-1/x)(some other value)=0

Why?
if n=3, we get (x-1/x)(x^2 + 1/(x^2) + 1) = 0. (x^2 + 1/(x^2) + 1) can never be equal to zero. Extending the same logic in all cases

if x = 1/x then x = either +1 or -1

statement 1 - x is an integer
no use to me since both +1 and -1 are integers

statement 2, n ≠ 0
no use to me as no matter what value of n I put, x will always be either +1 or -1

Hence both statements are insufficient

bumpbot · Answer

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

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

Prep Toolkit

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