Determine whether each of the following is positive or negative. $-$ $x^{n}$ when $n$ is odd and $x$ is negative $-$ $x^{n}$ when $n$ is even and $x$ is negative $x^{n}$ when $n$ is odd and $x$ is positive $x^{n}$ when $n$ is even and $x$ is positive
Solution
Step 1 ：Determine whether each of the following is positive or negative.
Step 2 ：For \(x^{n}\) when \(n\) is odd and \(x\) is negative, the result is \(\boxed{\text{Negative}}\).
Step 3 ：For \(x^{n}\) when \(n\) is even and \(x\) is negative, the result is \(\boxed{\text{Positive}}\).
Step 4 ：For \(x^{n}\) when \(n\) is odd and \(x\) is positive, the result is \(\boxed{\text{Positive}}\).
Step 5 ：For \(x^{n}\) when \(n\) is even and \(x\) is positive, the result is \(\boxed{\text{Positive}}\).
Step 6 ：The sign of \(x^n\) depends on both the sign of \(x\) and the parity of \(n\). If \(x\) is positive, \(x^n\) is always positive, regardless of whether \(n\) is odd or even. If \(x\) is negative, \(x^n\) is negative when \(n\) is odd and positive when \(n\) is even. This is because an even number of negative factors will always multiply to give a positive product, while an odd number of negative factors will always multiply to give a negative product.
