1. **State the problem:** Simplify the expression $$\frac{(x^{-2} \cdot x^{4})^{3}}{(x^{5} \cdot x^{-3})^{-2}}$$. 2. **Recall the exponent rules:** - Product of powers: $$a^{m} \cdot a^{n} = a^{m+n}$$ - Power of a power: $$(a^{m})^{n} = a^{m \cdot n}$$ - Quotient of powers: $$\frac{a^{m}}{a^{n}} = a^{m-n}$$ - Negative exponent: $$a^{-m} = \frac{1}{a^{m}}$$ 3. **Simplify numerator:** $$ (x^{-2} \cdot x^{4})^{3} = (x^{-2+4})^{3} = (x^{2})^{3} = x^{2 \cdot 3} = x^{6} $$ 4. **Simplify denominator:** $$ (x^{5} \cdot x^{-3})^{-2} = (x^{5-3})^{-2} = (x^{2})^{-2} = x^{2 \cdot (-2)} = x^{-4} $$ 5. **Combine numerator and denominator:** $$ \frac{x^{6}}{x^{-4}} = x^{6 - (-4)} = x^{6 + 4} = x^{10} $$ **Final answer:** $$x^{10}$$