1. **Stating the problem:** Calculate the value of the expression: $$\left[1,3^3 \cdot \left(\frac{3}{4}\right)^{-2}\right]^6 : \left[\left(\frac{16}{12}\right)^{10} : \left(\frac{3}{4}\right)^5\right]^2$$ 2. **Recall the rules and formulas:** - Negative exponents: $a^{-n} = \frac{1}{a^n}$ - Power of a product: $(ab)^n = a^n b^n$ - Power of a quotient: $\left(\frac{a}{b}\right)^n = \frac{a^n}{b^n}$ - Division of powers: $\frac{a^m}{a^n} = a^{m-n}$ - When dividing expressions with exponents, subtract exponents if bases are the same. 3. **Simplify inside the first bracket:** Calculate $3^3 = 27$ Calculate $\left(\frac{3}{4}\right)^{-2} = \left(\frac{4}{3}\right)^2 = \frac{16}{9}$ Multiply: $27 \cdot \frac{16}{9} = \frac{27 \cdot 16}{9} = \frac{432}{9} = 48$ So, first bracket is $[1, 48]^6 = 48^6$ 4. **Simplify inside the second bracket:** Calculate $\left(\frac{16}{12}\right)^{10} = \left(\frac{4}{3}\right)^{10}$ Calculate $\left(\frac{3}{4}\right)^5$ Divide: $\frac{\left(\frac{4}{3}\right)^{10}}{\left(\frac{3}{4}\right)^5} = \left(\frac{4}{3}\right)^{10} \cdot \left(\frac{4}{3}\right)^5 = \left(\frac{4}{3}\right)^{15}$ 5. **Raise the second bracket to the power 2:** $$\left[\left(\frac{4}{3}\right)^{15}\right]^2 = \left(\frac{4}{3}\right)^{30}$$ 6. **Divide the two results:** $$\frac{48^6}{\left(\frac{4}{3}\right)^{30}}$$ Rewrite $48$ as $48 = 16 \cdot 3 = (4^2) \cdot 3$ So, $$48^6 = \left(4^2 \cdot 3\right)^6 = 4^{12} \cdot 3^6$$ Rewrite denominator: $$\left(\frac{4}{3}\right)^{30} = \frac{4^{30}}{3^{30}}$$ Divide: $$\frac{4^{12} \cdot 3^6}{\frac{4^{30}}{3^{30}}} = 4^{12} \cdot 3^6 \cdot \frac{3^{30}}{4^{30}} = 4^{12 - 30} \cdot 3^{6 + 30} = 4^{-18} \cdot 3^{36}$$ Rewrite $4^{-18} = \frac{1}{4^{18}} = \frac{1}{(2^2)^{18}} = \frac{1}{2^{36}}$ Rewrite $3^{36}$ as is. So the expression equals: $$\frac{3^{36}}{2^{36}} = \left(\frac{3}{2}\right)^{36}$$ **Final answer:** $$\boxed{\left(\frac{3}{2}\right)^{36}}$$