Which expression is equivalent to \( \frac{28 p^{9} q^{-5}}{12 p^{-6} q^{7}} \) ? Assume \( p \neq 0, q \neq 0 \) \( \frac{2}{p^{15} q^{12}} \) \( \frac{7 p^{15}}{3 q^{12}} \) \( \frac{2 q^{12}}{p^{15}} \) \( \frac{7 p^{15} q^{12}}{3} \)

To simplify the expression \( \frac{28 p^{9} q^{-5}}{12 p^{-6} q^{7}} \), we first simplify the coefficients: \( \frac{28}{12} = \frac{7}{3} \). Next, we handle the variables \( p \) and \( q \) using the laws of exponents: - For \( p \): \( p^{9} \div p^{-6} = p^{9 - (-6)} = p^{9 + 6} = p^{15} \) - For \( q \): \( q^{-5} \div q^{7} = q^{-5 - 7} = q^{-12} \) Putting it all together, we have: \[ \frac{28 p^{9} q^{-5}}{12 p^{-6} q^{7}} = \frac{7 p^{15} q^{-12}}{3} = \frac{7 p^{15}}{3 q^{12}} \] Thus, the expression simplifies to \( \frac{7 p^{15}}{3 q^{12}} \). And there's your answer! Now, how about diving into some fun facts? Understanding the rules of exponents can prevent you from getting tangled up in algebraic expressions like this one. Remember, when you divide like bases, you subtract the exponents! It's a simple yet powerful tool that makes handling polynomials and rational expressions a breeze. Additionally, this simplification process is not just reserved for algebraic exercises—it's often used in fields like physics and engineering, where you might encounter complex equations and need to reduce them to more manageable forms for calculations. Knowing how to quickly simplify can save you time and effort in problem-solving!