Let's analyze the given problem: If [tex]\(\frac{n^x}{n^y}=n^2\)[/tex] for all [tex]\(n \neq 0\)[/tex], which of the options must be true?
We start with the equation:
[tex]\[
\frac{n^x}{n^y} = n^2
\][/tex]
We'll use properties of exponents to simplify the left side of the equation. Recall that [tex]\(\frac{n^x}{n^y} = n^{x-y}\)[/tex]. Thus, we can rewrite the equation as:
[tex]\[
n^{x-y} = n^2
\][/tex]
Given that the bases (both [tex]\(n\)[/tex]) are the same and non-zero, we equate the exponents on both sides of the equation. This gives us:
[tex]\[
x - y = 2
\][/tex]
Therefore, the correct statement is:
[tex]\[
\boxed{x - y = 2}
\][/tex]
So, the answer is:
[tex]\[
G. \; x - y = 2
\][/tex]
