1. **State the problem:** Simplify the expression $$\left\{\left[\left(\frac{1}{4}\right)^3 \cdot \left(\frac{2}{3}\right)^3\right]^{-1} \cdot \left(\frac{1}{6}\right)^4\right\}^{-1} : \left[\left(\frac{1}{4}\right)^3 \cdot \left(\frac{2}{5}\right)^3\right]^0$$. 2. **Recall important rules:** - Any number to the zero power is 1: $$a^0 = 1$$ for $$a \neq 0$$. - Negative exponents mean reciprocal: $$a^{-n} = \frac{1}{a^n}$$. - Power of a product: $$(ab)^n = a^n b^n$$. - Division of expressions: $$\frac{A}{B} = A \cdot B^{-1}$$. 3. **Simplify the right side first:** $$\left[\left(\frac{1}{4}\right)^3 \cdot \left(\frac{2}{5}\right)^3\right]^0 = 1$$ 4. **Simplify the left side inside the curly braces:** $$\left[\left(\frac{1}{4}\right)^3 \cdot \left(\frac{2}{3}\right)^3\right]^{-1} = \left(\frac{1}{4} \cdot \frac{2}{3}\right)^{-3} = \left(\frac{1 \cdot 2}{4 \cdot 3}\right)^{-3} = \left(\frac{2}{12}\right)^{-3} = \left(\frac{1}{6}\right)^{-3}$$ 5. **Multiply by $$\left(\frac{1}{6}\right)^4$$:** $$\left(\frac{1}{6}\right)^{-3} \cdot \left(\frac{1}{6}\right)^4 = \left(\frac{1}{6}\right)^{-3+4} = \left(\frac{1}{6}\right)^1 = \frac{1}{6}$$ 6. **Apply the outer negative exponent:** $$\left(\frac{1}{6}\right)^{-1} = 6$$ 7. **Divide by the right side (which is 1):** $$6 : 1 = 6$$ **Final answer:** $$6$$