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
Given: n is an integer and \(x^n – x^{-n} = 0\) In other words: \(x^n = x^{-n} \)
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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 \(x \neq 0\), \(x \neq 1\) and \(x \neq -1\)
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 \(m=n\)
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So, if \(x^n = x^{-n}\), then there are 4 possible cases:
case a: \(x \neq 0\), \(x \neq 1\) and \(x \neq -1\), 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 integerTarget 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 = 1Case b: x = -1 and n = 2. In this case, the answer to the target question is
x = -1Since we can’t answer the
target question with certainty, statement 1 is NOT SUFFICIENT
Statement 2: n ≠ 0PRO 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 = 1Case b: x = -1 and n = 2. In this case, the answer to the target question is
x = -1Since 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 = 1Case b: x = -1 and n = 2. In this case, the answer to the target question is
x = -1Since we still can’t answer the
target question with certainty, the combined statements are NOT SUFFICIENT
Answer: E
Cheers,
Brent
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