1. Problem: Simplify the expression $\left(2x^3 y^{-3}\right)^{-2}$. Formula: When raising a power to another power, multiply the exponents: $\left(a^m\right)^n = a^{m \cdot n}$. Step 1: Apply the power to each factor inside the parentheses: $$\left(2x^3 y^{-3}\right)^{-2} = 2^{-2} \cdot \left(x^3\right)^{-2} \cdot \left(y^{-3}\right)^{-2}$$ Step 2: Simplify each term: $$2^{-2} = \frac{1}{2^2} = \frac{1}{4}$$ $$\left(x^3\right)^{-2} = x^{3 \cdot (-2)} = x^{-6}$$ $$\left(y^{-3}\right)^{-2} = y^{-3 \cdot (-2)} = y^6$$ Step 3: Combine the terms: $$\frac{1}{4} x^{-6} y^6 = \frac{y^6}{4 x^6}$$ --- 2. Problem: Simplify the expression $5x^2 y \cdot (2x^4 y^{-3})$. Formula: Multiply coefficients and add exponents of like bases: $a^m \cdot a^n = a^{m+n}$. Step 1: Multiply coefficients: $$5 \cdot 2 = 10$$ Step 2: Add exponents for $x$: $$x^{2+4} = x^6$$ Step 3: Add exponents for $y$: $$y^{1 + (-3)} = y^{-2}$$ Step 4: Combine: $$10 x^6 y^{-2} = \frac{10 x^6}{y^2}$$ --- 3. Problem: Simplify $\left( \frac{-7 a^2 b^3 c^0}{3 a^3 b^4 c^3} \right)^{-4}$. Note: $c^0 = 1$. Step 1: Simplify inside the parentheses: $$\frac{-7}{3} \cdot a^{2-3} \cdot b^{3-4} \cdot c^{0-3} = \frac{-7}{3} a^{-1} b^{-1} c^{-3}$$ Step 2: Apply the negative fourth power: $$\left( \frac{-7}{3} a^{-1} b^{-1} c^{-3} \right)^{-4} = \left( \frac{-7}{3} \right)^{-4} a^{-1 \cdot (-4)} b^{-1 \cdot (-4)} c^{-3 \cdot (-4)}$$ Step 3: Simplify each term: $$\left( \frac{-7}{3} \right)^{-4} = \left( \frac{3}{-7} \right)^4 = \frac{3^4}{(-7)^4} = \frac{81}{2401}$$ $$a^{4}$$ $$b^{4}$$ $$c^{12}$$ Step 4: Combine: $$\frac{81}{2401} a^4 b^4 c^{12}$$ --- 4. Problem: Simplify $\left( \frac{-2 a^3 b^2 c^0}{3 a^2 b^3 c^7} \right)^{-2}$. Note: $c^0 = 1$. Step 1: Simplify inside the parentheses: $$\frac{-2}{3} \cdot a^{3-2} \cdot b^{2-3} \cdot c^{0-7} = \frac{-2}{3} a^{1} b^{-1} c^{-7}$$ Step 2: Apply the negative second power: $$\left( \frac{-2}{3} a^{1} b^{-1} c^{-7} \right)^{-2} = \left( \frac{-2}{3} \right)^{-2} a^{1 \cdot (-2)} b^{-1 \cdot (-2)} c^{-7 \cdot (-2)}$$ Step 3: Simplify each term: $$\left( \frac{-2}{3} \right)^{-2} = \left( \frac{3}{-2} \right)^2 = \frac{9}{4}$$ $$a^{-2}$$ $$b^{2}$$ $$c^{14}$$ Step 4: Combine: $$\frac{9}{4} a^{-2} b^{2} c^{14} = \frac{9 b^{2} c^{14}}{4 a^{2}}$$