1. **Problem statement:** Simplify the expression $$\frac{\sqrt{\frac{a^{13}b^{4}}{(d^{3}c)^{3}}}}{\sqrt{\frac{a^{9}c^{5}}{d^{x}}}} = \frac{a^{2}b^{2}d^{2}}{c^{4}}$$ and find the value of $x$ by comparing terms if possible. 2. **Rewrite the radicals:** Recall that $\sqrt{\frac{A}{B}} = \frac{\sqrt{A}}{\sqrt{B}}$ and $\sqrt{M^{n}} = M^{\frac{n}{2}}$. First, simplify inside the radicals: $$\sqrt{\frac{a^{13}b^{4}}{(d^{3}c)^{3}}} = \sqrt{\frac{a^{13}b^{4}}{d^{9}c^{3}}} = \frac{a^{\frac{13}{2}}b^{2}}{d^{\frac{9}{2}}c^{\frac{3}{2}}}$$ Similarly, $$\sqrt{\frac{a^{9}c^{5}}{d^{x}}} = \frac{a^{\frac{9}{2}}c^{\frac{5}{2}}}{d^{\frac{x}{2}}}$$ 3. **Form the division:** $$\frac{\frac{a^{\frac{13}{2}}b^{2}}{d^{\frac{9}{2}}c^{\frac{3}{2}}}}{\frac{a^{\frac{9}{2}}c^{\frac{5}{2}}}{d^{\frac{x}{2}}}} = \frac{a^{\frac{13}{2}}b^{2}}{d^{\frac{9}{2}}c^{\frac{3}{2}}} \times \frac{d^{\frac{x}{2}}}{a^{\frac{9}{2}}c^{\frac{5}{2}}} = \frac{a^{\frac{13}{2} - \frac{9}{2}} b^{2} d^{\frac{x}{2} - \frac{9}{2}}}{c^{\frac{3}{2} + \frac{5}{2}}}$$ Simplify exponents: $$a^{2} b^{2} d^{\frac{x - 9}{2}} c^{-4}$$ 4. **Set equal to the right side:** $$\frac{a^{2} b^{2} d^{\frac{x - 9}{2}}}{c^{4}} = \frac{a^{2} b^{2} d^{2}}{c^{4}}$$ 5. **Compare terms:** Since bases $a,b,c$ are nonzero and non-one, equate exponents: For $a$: $2 = 2$ (true) For $b$: $2 = 2$ (true) For $c$: $-4 = -4$ (true) For $d$: $\frac{x - 9}{2} = 2$ 6. **Solve for $x$:** $$\frac{x - 9}{2} = 2$$ Multiply both sides by 2: $$\cancel{2} \times \frac{x - 9}{\cancel{2}} = 2 \times 2$$ $$x - 9 = 4$$ Add 9 to both sides: $$x = 13$$ 7. **Conclusion:** The equation is solvable and the solution is $$x = 13$$