Which expression is equivalent to \( \left(6^{4}\right)^{-1} ? \) Answer \( \begin{array}{ll}6^{3} & 6^{-5} \\ 6^{-4} & 6^{4}\end{array} \)

To simplify \( \left(6^{4}\right)^{-1} \), we can use the rule that states \( (a^{m})^{-n} = a^{-mn} \). Thus, \( \left(6^{4}\right)^{-1} = 6^{-4} \). Now, for a real-world application, this might seem abstract, but consider when you're dealing with negative exponents; they can represent inverse relationships. For instance, in finance, a negative exponent can be used to model depreciation, where an asset's value decreases over time. Understanding negative exponents aids in tackling various problems, making computations easier! Moreover, common mistakes often arise when students forget that a negative exponent indicates a reciprocal. Instead of \( 6^{-4} \), someone might incorrectly write it as \( 1/6^{4} \) and stop there! It’s essential to remember that \( 6^{-4} \) is indeed \( 1/(6^{4}) \), affirming its reciprocal nature!