- \( x^{-4}=1 \)

Did you know that equations like \( x^{-4} = 1 \) can lead us to discover something intriguing about exponents? When you rewrite \( x^{-4} \) as \( \frac{1}{x^4} = 1 \), it becomes clear that we are looking for values of \( x \) where the reciprocal of \( x^4 \) equals 1. This means \( x^4 = 1\) which leads us to the exciting solutions of \( x = 1 \) and \( x = -1 \)! Now, let's talk about finding solutions to equations like this. A common mistake is forgetting that there can be multiple solutions, especially when dealing with even roots or exponents. In this case, since we're dealing with a fourth power, don’t forget to check both the positive and negative roots. So remember, you might just stumble upon multiple answers hiding in plain sight!