(10) \( \frac{(2 a)^{-3}}{16 a^{-4}} \)

1. Rewrite the numerator: \[ (2a)^{-3} = 2^{-3} \cdot a^{-3} = \frac{1}{8}a^{-3} \] 2. Write the expression with the simplified numerator: \[ \frac{(2a)^{-3}}{16a^{-4}} = \frac{\frac{1}{8}a^{-3}}{16a^{-4}} \] 3. Replace the division by multiplication using the reciprocal of the denominator: \[ \frac{\frac{1}{8}a^{-3}}{16a^{-4}} = \frac{1}{8}a^{-3} \times \frac{1}{16a^{-4}}^{-1} \] Note that taking the reciprocal of \(16a^{-4}\) gives: \[ \frac{1}{16a^{-4}} = \frac{a^4}{16} \] So the expression becomes: \[ \frac{1}{8}a^{-3} \times \frac{a^4}{16} \] 4. Multiply the constants and combine the powers of \(a\): \[ \frac{1}{8} \times \frac{1}{16} = \frac{1}{128} \] and \[ a^{-3} \times a^{4} = a^{-3+4} = a^{1} = a \] 5. Thus, the simplified expression is: \[ \frac{a}{128} \]