91Ó°ÊÓ

We need to show that if \(x \neq 0\), then \(|x^n| = |x|^n\) for all integers \(n\). By considering different cases for \(n\) (positive, negative, and 0), we demonstrated that the statement holds true for each case: 1. For positive values of \(n\), raising the absolute values to the power of \(n\) maintains the non-negative properties, so the equation holds. 2. For negative values of \(n\), by using the property \(x^{-n} = \frac{1}{x^n}\), we reduced the equation to a non-negative form, keeping the equation true. 3. For \(n = 0\), any non-zero number raised to the power of 0 is 1, so the equation is true in this case as well. Thus, we conclude that if \(x \neq 0\), then \(|x^n| = |x|^n\) for all integers \(n\).