1. **State the problem:** Simplify the expression $3p^6q^{-4} \cdot 5p^{-8}q^7$. 2. **Recall the rules:** When multiplying terms with the same base, add the exponents: $a^m \cdot a^n = a^{m+n}$. 3. **Multiply the coefficients:** $3 \cdot 5 = 15$. 4. **Add the exponents for $p$:** $p^{6} \cdot p^{-8} = p^{6 + (-8)} = p^{-2}$. 5. **Add the exponents for $q$:** $q^{-4} \cdot q^{7} = q^{-4 + 7} = q^{3}$. 6. **Combine all parts:** $$15p^{-2}q^{3}$$. 7. **Rewrite negative exponent:** $p^{-2} = \frac{1}{p^{2}}$. 8. **Final simplified expression:** $$\frac{15q^{3}}{p^{2}}$$.