1. Simplify the expression $ \frac{12 a^0 b^3 c^{-1}}{15 a^{-2} b^5 c^{-3}} $. 2. Simplify the expression $ \frac{(3 x y^{-2})^{-2}}{3 x^{-2} y} $. ### Step 1: Simplify $ \frac{12 a^0 b^3 c^{-1}}{15 a^{-2} b^5 c^{-3}} $ 1. Recall that $a^0 = 1$ for any $a \neq 0$. 2. Rewrite the expression: $$ \frac{12 \cdot 1 \cdot b^3 \cdot c^{-1}}{15 \cdot a^{-2} \cdot b^5 \cdot c^{-3}} = \frac{12 b^3 c^{-1}}{15 a^{-2} b^5 c^{-3}} $$ 3. Apply the rule $ \frac{x^m}{x^n} = x^{m-n} $ to each variable: $$ = \frac{12}{15} \cdot a^{0 - (-2)} \cdot b^{3 - 5} \cdot c^{-1 - (-3)} = \frac{12}{15} \cdot a^{2} \cdot b^{-2} \cdot c^{2} $$ 4. Simplify the fraction $ \frac{12}{15} $: $$ \frac{\cancel{12}}{\cancel{15}} = \frac{4}{5} $$ 5. Final simplified form: $$ \frac{4}{5} a^{2} b^{-2} c^{2} = \frac{4 a^{2} c^{2}}{5 b^{2}} $$ --- ### Step 2: Simplify $ \frac{(3 x y^{-2})^{-2}}{3 x^{-2} y} $ 1. Apply the power of a product rule $ (abc)^n = a^n b^n c^n $: $$ (3 x y^{-2})^{-2} = 3^{-2} x^{-2} (y^{-2})^{-2} = 3^{-2} x^{-2} y^{4} $$ 2. Substitute back into the expression: $$ \frac{3^{-2} x^{-2} y^{4}}{3 x^{-2} y} $$ 3. Simplify the fraction of constants: $$ \frac{3^{-2}}{3} = \frac{\frac{1}{3^{2}}}{3} = \frac{1}{3^{2} \cdot 3} = \frac{1}{3^{3}} = 3^{-3} $$ 4. Simplify the variables using $ \frac{x^m}{x^n} = x^{m-n} $: $$ x^{-2 - (-2)} = x^{0} = 1 $$ $$ y^{4 - 1} = y^{3} $$ 5. Combine all parts: $$ 3^{-3} \cdot 1 \cdot y^{3} = \frac{y^{3}}{3^{3}} = \frac{y^{3}}{27} $$ --- ### Final answers: 1. $ \frac{4 a^{2} c^{2}}{5 b^{2}} $ 2. $ \frac{y^{3}}{27} $