What happens when we multiply \((-1)\) to itself multiple times? There is a pattern to find that makes simplifying exponent problems with a negative base much more simple.
The basic rule when a negative number is raised to an exponent \(({-}{\text{b}})^{\text{x}}\):
If the power is even → the answer is positive,
If the power is odd → the answer is negative.
It is important to know that \(-{\text{b}}^{\text{x}}\) is different than \(({-}{\text{b}})^{\text{x}}\). When you have \(-{\text{b}}^{\text{x}}\) , where the negative is not inside the parentheses, the exponent does not apply to it. This is because of the order of operations.
In our previous examples, the negative was inside the parentheses, so it was being raised to the exponent as well. If there aren’t parentheses, the exponent doesn’t apply to it. This is due to the order of operations.
