1. Simplify \(\left(3x^2 y^3\right)^2\): Step 1: Apply the power to each factor inside the parentheses: $$\left(3\right)^2 \left(x^2\right)^2 \left(y^3\right)^2 = 3^2 \times x^{2 \times 2} \times y^{3 \times 2}$$ Step 2: Calculate powers: $$= 9 x^4 y^6$$ 2. Simplify \(\left(4x^2 yz^3\right)^2 \left(2xy^2 z\right)^2\): Step 1: Simplify each term inside parentheses: $$\left(4\right)^2 \left(x^2\right)^2 y^2 \left(z^3\right)^2 = 16 x^4 y^2 z^6$$ $$\left(2\right)^2 x^2 y^{2 \times 2} z^2 = 4 x^2 y^4 z^2$$ Step 2: Multiply the two results by adding exponents for like bases: $$16 \times 4 = 64$$ $$x^{4+2} = x^6$$ $$y^{2+4} = y^6$$ $$z^{6+2} = z^8$$ Step 3: Final simplification: $$64 x^6 y^6 z^8$$ 3. Simplify \(\left(4xy^2 z^3\right)^3\): Step 1: Apply power 3 to each factor: $$4^3 x^3 y^{2 \times 3} z^{3 \times 3} = 64 x^3 y^6 z^9$$ 4. Simplify \(\left(3xy^2 z^4\right)^2 \left(-3x^2 y^3 z\right)^3\): Step 1: Simplify each power: $$\left(3\right)^2 x^2 y^4 z^8 = 9 x^2 y^4 z^8$$ $$\left(-3\right)^3 x^{2 \times 3} y^{3 \times 3} z^3 = -27 x^6 y^9 z^3$$ Step 2: Multiply the simplified forms: $$9 \times -27 = -243$$ $$x^{2+6} = x^8$$ $$y^{4+9} = y^{13}$$ $$z^{8+3} = z^{11}$$ Step 3: Final answer: $$-243 x^8 y^{13} z^{11}$$ --- Solve for the unknown(s): 1. Solve \(\frac{7}{2} : x = \frac{4}{5} : 3\): Step 1: Write the proportion as \(\frac{7/2}{x} = \frac{4/5}{3}\). Step 2: Simplify right side: $$\frac{4/5}{3} = \frac{4/5}{3/1} = \frac{4}{5} \times \frac{1}{3} = \frac{4}{15}$$ Step 3: Set equation: $$\frac{7/2}{x} = \frac{4}{15} \implies \frac{7}{2x} = \frac{4}{15}$$ Step 4: Cross-multiply: $$7 \times 15 = 4 \times 2x \implies 105 = 8x$$ Step 5: Solve for \(x\): $$x = \frac{105}{8} = 13.125$$ 2. Solve \((x-1) : (2x+1) = 4 : 11\): Step 1: Write as fraction equalities: $$\frac{x-1}{2x+1} = \frac{4}{11}$$ Step 2: Cross-multiply: $$11(x-1) = 4(2x+1)$$ Step 3: Expand: $$11x - 11 = 8x + 4$$ Step 4: Rearrange terms: $$11x - 8x = 4 + 11$$ $$3x = 15$$ Step 5: Solve for \(x\): $$x = 5$$ 3. Solve \(\frac{5}{9} : \frac{1}{2} = x : \frac{3}{4}\): Step 1: Write as proportion: $$\frac{5/9}{1/2} = \frac{x}{3/4}$$ Step 2: Simplify left side: $$\frac{5/9}{1/2} = \frac{5}{9} \times \frac{2}{1} = \frac{10}{9}$$ Step 3: Equation: $$\frac{10}{9} = \frac{x}{3/4}$$ Step 4: Solve for \(x\): $$x = \frac{10}{9} \times \frac{3}{4} = \frac{30}{36} = \frac{5}{6}$$ 4. Solve \((2x - 1) : (x + 1) = 7 : 5\): Step 1: Write as fraction equality: $$\frac{2x - 1}{x + 1} = \frac{7}{5}$$ Step 2: Cross multiply: $$5(2x -1) = 7(x + 1)$$ Step 3: Expand: $$10x - 5 = 7x + 7$$ Step 4: Rearrange: $$10x - 7x = 7 + 5$$ $$3x = 12$$ Step 5: Solve for \(x\): $$x = 4$$