1. **State the problem:** Simplify the expression $$\frac{5p^{4} \times (3q^{3})^{2}}{15p^{6}q^{4}}$$. 2. **Apply the exponent rule:** Recall that $ (a^m)^n = a^{m \times n} $. So, $ (3q^{3})^{2} = 3^{2} \times (q^{3})^{2} = 9q^{6} $. 3. **Rewrite the expression:** $$\frac{5p^{4} \times 9q^{6}}{15p^{6}q^{4}} = \frac{45p^{4}q^{6}}{15p^{6}q^{4}}$$ 4. **Simplify the fraction coefficients:** $$\frac{\cancel{45}^{3}p^{4}q^{6}}{\cancel{15}^{1}p^{6}q^{4}} = \frac{3p^{4}q^{6}}{p^{6}q^{4}}$$ 5. **Simplify the variables using the quotient rule $\frac{a^{m}}{a^{n}} = a^{m-n}$:** $$3p^{4-6}q^{6-4} = 3p^{-2}q^{2}$$ 6. **Rewrite negative exponents as positive by moving to denominator:** $$3 \times \frac{q^{2}}{p^{2}} = \frac{3q^{2}}{p^{2}}$$ **Final answer:** $$\frac{3q^{2}}{p^{2}}$$