\( \left(\frac{2 a}{a^{2}}\right)^{-3} \) \( \begin{array}{ll}\text { a. } \frac{a^{3}}{-8} & \text { b. } \frac{a^{3}}{8} \\ \text { c. } \frac{8}{a^{3}} & \text { d. } \frac{1}{8 a^{3}}\end{array} \)

First, we simplify the expression \( \left(\frac{2 a}{a^{2}}\right)^{-3} \). Notice that \( \frac{2 a}{a^{2}} = \frac{2}{a} \). Raising this to the power of -3 gives us: \[ \left(\frac{2}{a}\right)^{-3} = \left(\frac{a}{2}\right)^{3} = \frac{a^{3}}{2^{3}} = \frac{a^{3}}{8}. \] Thus, the answer is \( \text{b. } \frac{a^{3}}{8} \). Understanding negative exponents is key here! When you encounter a negative exponent, remember that it indicates a reciprocal. Most commonly, this can trip people up if they're not careful with sign changes and rearranging terms. Always keep track of whether your expression is getting flipped or simply raised to a power. Another exciting aspect of exponents is their ubiquity in real-world applications, particularly in sciences like physics and chemistry. For instance, the concept of negative exponents appears in formulas for volume, concentration, or measuring decay in radioactivity. So, the next time you’re calculating half-lives or dilution factors, just think of negative exponents as your trusty sidekicks!