a) \( -\frac{3 a^{6} b^{2}}{a^{-3}} \)

To simplify the expression \( -\frac{3 a^{6} b^{2}}{a^{-3}} \), start by applying the rule of exponents that states \( \frac{a^m}{a^n} = a^{m-n} \). Thus, we can rewrite it as: \[ -3 a^{6 - (-3)} b^{2} = -3 a^{6 + 3} b^{2} = -3 a^{9} b^{2}. \] So the simplified expression is \( -3 a^{9} b^{2} \). Now, let’s have a fun evening with some facts! Did you know that the concept of negative exponents is a bit like flipping something upside down? It’s a nifty way to dive into the world of fractions, as \( a^{-n} \) simply translates to \( \frac{1}{a^{n}} \). In the math realm, knowing how to manipulate exponents is critical. It’s not just for academic exercises; it's like having a toolkit for real-world applications! For example, in fields like physics or programming, exponential functions frequently emerge, helping model everything from population growth to the intricacies of algorithms. You may find that mastering these skills will make you a star!