1. **State the problem:** Simplify the expression $$\frac{9^{x-1} \cdot 25^x}{5^{x-2} \cdot 15^{x+2} \cdot 3^x}$$. 2. **Rewrite bases as prime factors:** - $9 = 3^2$ - $25 = 5^2$ - $15 = 3 \cdot 5$ So the expression becomes: $$\frac{(3^2)^{x-1} \cdot (5^2)^x}{5^{x-2} \cdot (3 \cdot 5)^{x+2} \cdot 3^x}$$ 3. **Apply power of a power rule:** $$\frac{3^{2(x-1)} \cdot 5^{2x}}{5^{x-2} \cdot 3^{x+2} \cdot 5^{x+2} \cdot 3^x}$$ 4. **Combine like bases in denominator:** $$5^{x-2} \cdot 5^{x+2} = 5^{(x-2)+(x+2)} = 5^{2x}$$ Similarly for base 3 in denominator: $$3^{x+2} \cdot 3^x = 3^{(x+2)+x} = 3^{2x+2}$$ So denominator is: $$5^{2x} \cdot 3^{2x+2}$$ 5. **Rewrite entire expression:** $$\frac{3^{2x-2} \cdot 5^{2x}}{5^{2x} \cdot 3^{2x+2}}$$ 6. **Cancel common terms:** - $5^{2x}$ cancels out. 7. **Simplify powers of 3:** $$3^{2x-2} \div 3^{2x+2} = 3^{(2x-2)-(2x+2)} = 3^{-4}$$ 8. **Final simplified expression:** $$3^{-4} = \frac{1}{3^4} = \frac{1}{81}$$ **Answer:** The simplified form of the expression is $$\frac{1}{81}$$.