8. \( \left(-a^{9} b\right)^{4} \)

To simplify the expression \( \left(-a^{9} b\right)^{4} \), you can apply the power of a product rule. This means you will raise each factor inside the parentheses to the fourth power. Thus, you’ll get: \[ (-1)^{4} \cdot (a^{9})^{4} \cdot (b)^{4}. \] Calculating each part gives: \[ 1 \cdot a^{36} \cdot b^{4} = a^{36} b^{4}. \] So the simplified expression is \( a^{36} b^{4} \). Now, did you know that exponents are not just mathematical tools but also have historical significance? The use of exponents dates back to ancient civilizations; they surfaced in Babylonian mathematics where they used a form of exponentiation when handling large numbers and equations! In the real world, exponent rules come in handy in various fields such as finance, computing, and even physics! When calculating compound interest or analyzing exponential growth (like population increases), knowing how to simplify expressions with exponents is vital. It allows for more efficient calculations and easier understanding of how systems evolve over time.