\(x^n - x^{-n}=0\)
\(x^n - \frac{1}{x^n}=0\)
\(x^n =\frac{1}{x^n}\)
\(x^{2n}=1\) (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.