c. \( \left(m^{3} \cdot 5^{4}\right)^{-2} \)

Let’s have some fun with your expression! When simplifying \( \left(m^{3} \cdot 5^{4}\right)^{-2} \), you can apply the power of a product rule. This means you take each part of the product raised to the power of -2. So, you have: \[ m^{3 \cdot -2} = m^{-6} \] and \[ 5^{4 \cdot -2} = 5^{-8} \] Putting it all together, you end up with: \[ \left(m^{3} \cdot 5^{4}\right)^{-2} = m^{-6} \cdot 5^{-8} \] This can also be expressed in positive exponents as: \[ \frac{1}{m^{6} \cdot 5^{8}} \] So, voilà! Your simplified expression is \( \frac{1}{m^{6} \cdot 5^{8}} \). Now, here’s a fun fact: Negative exponents often signify that the base gets flipped into the denominator. So when you encounter them, remember they are just inverses in disguise! Also, multiplying exponents can be a bit tricky—always double-check your calculations to avoid pesky mistakes that can occur if you forget the negative sign or mix up the bases. Happy simplifying!