1. Simplify \(\frac{4-\frac{9}{x^2}}{2-\frac{3}{x}}\). Rewrite numerator: \(4-\frac{9}{x^2} = \frac{4x^2 - 9}{x^2} = \frac{(2x - 3)(2x + 3)}{x^2}\). Rewrite denominator: \(2-\frac{3}{x} = \frac{2x - 3}{x}\). Divide numerator by denominator: $$\frac{\frac{(2x - 3)(2x + 3)}{x^2}}{\frac{2x - 3}{x}} = \frac{(2x - 3)(2x + 3)}{x^2} \times \frac{x}{2x - 3} = \frac{2x + 3}{x} = 2 + \frac{3}{x}.$$ Answer: (a) \(2 + \frac{3}{x}\). 2. Simplify \(\frac{\frac{1}{x} - \frac{1}{y}}{x - y}\). The numerator: \(\frac{1}{x} - \frac{1}{y} = \frac{y - x}{xy} = -\frac{x - y}{xy}\). Divide by \(x-y\): $$\frac{-\frac{x - y}{xy}}{x - y} = -\frac{x - y}{xy} \times \frac{1}{x - y} = -\frac{1}{xy}.$$ Answer: (a) \(-\frac{1}{xy}\). 3. Simplify \(\frac{\frac{x}{y} - \frac{y}{x}}{\frac{1}{y} - \frac{1}{x}}\). Numerator: \(\frac{x}{y} - \frac{y}{x} = \frac{x^2 - y^2}{xy} = \frac{(x - y)(x + y)}{xy}\). Denominator: \(\frac{1}{y} - \frac{1}{x} = \frac{x - y}{xy}\). Divide: $$\frac{\frac{(x - y)(x + y)}{xy}}{\frac{x - y}{xy}} = \frac{(x - y)(x + y)}{xy} \times \frac{xy}{x - y} = x + y.$$ Answer: (b) \(x + y\). 4. Simplify \(\frac{\frac{x^2 - y^2}{xy}}{\frac{3}{x} - \frac{3}{y}}\). Numerator: \(\frac{x^2 - y^2}{xy} = \frac{(x - y)(x + y)}{xy}\). Denominator: \(\frac{3}{x} - \frac{3}{y} = 3\left(\frac{1}{x} - \frac{1}{y}\right) = 3 \frac{y - x}{xy} = -3 \frac{x - y}{xy}\). Divide: $$\frac{\frac{(x - y)(x + y)}{xy}}{-3 \frac{x - y}{xy}} = \frac{(x - y)(x + y)}{xy} \times \frac{xy}{-3(x - y)} = -\frac{x + y}{3}.$$ No answer matches exactly among choices (a), (b), (c), answer is (d) none. 5. Simplify \(\sqrt[3]{9x^5y^2} \cdot \sqrt[3]{9z^5} = \sqrt[3]{81 x^5 y^2 z^5}\). Since \(81 = 3^4\), rewrite inside cube root: \(3^4 x^{5} y^{2} z^{5} = 3^3 \times 3 x^{3} x^{2} y^{2} z^{3} z^{2}\). Separate perfect cubes: $$3^3 x^3 z^3 \times 3 x^2 y^2 z^2 = (3 x z)^3 \times 3 x^2 y^2 z^2.$$ Take cube root: $$3 x z \sqrt[3]{3 x^{2} y^{2} z^{2}}.$$ Answer: (c) \(3 x z \sqrt[3]{3 x^{2} y^{2} z^{2}}\). 6. Simplify \(\sqrt[4]{32 x^{15} y^{11}}\). Prime factors: \(32 = 2^{5}\), so inside is \(2^{5} x^{15} y^{11}\). Rewrite exponents in multiples of 4 plus remainder: \(2^{5} = 2^{4} \times 2^{1}\), \(x^{15} = x^{12} \times x^{3}\), \(y^{11} = y^{8} \times y^{3}\). Extract fourth roots: $$2^{4/4} x^{12/4} y^{8/4} \sqrt[4]{2^{1} x^{3} y^{3}} = 2 x^{3} y^{2} \sqrt[4]{2 x^{3} y^{3}}.$$ Answer: (c) \(2 x^{3} y^{2} \sqrt[4]{2 x^{3} y^{3}}\). 7. Simplify \(\frac{27 x^3 y^2}{\sqrt{9 x y}}\). Since \(\sqrt{9 x y} = 3 \sqrt{x y}\), write denominator as \(3 \sqrt{x y}\). Divide: $$\frac{27 x^3 y^2}{3 \sqrt{x y}} = 9 x^{3} y^{2} \frac{1}{\sqrt{x y}} = 9 x^{3} y^{2} \times (x y)^{-1/2} = 9 x^{3 - 1/2} y^{2 - 1/2} = 9 x^{5/2} y^{3/2}.$$ Express \(y^{3/2} = y \sqrt{y}\) to match answer: $$9 x^{5/2} y \sqrt{y} = 9 x^{5/2} y \sqrt{y}$$ Answer matches (a) \(9 x^{5/2} y\) only if ignoring \(\sqrt{y}\) factor, so no exact match. Check (b): \(9 x^{2} y \sqrt{x y} = 9 x^{2} y \sqrt{x y}\) expands as $$9 x^{2} y x^{1/2} y^{1/2} = 9 x^{2.5} y^{1.5} = 9 x^{5/2} y^{3/2},$$ perfect match. Answer: (b) \(9 x^{2} y \sqrt{x y}\). 8. Simplify \(\frac{\sqrt[3]{x^{4} y^{5}}}{\sqrt[3]{x y^{2}}}\). Divide inside cube roots: $$\sqrt[3]{\frac{x^{4} y^{5}}{x y^{2}}} = \sqrt[3]{x^{3} y^{3}} = \sqrt[3]{(x y)^3} = x y.$$ Answer: (a) \(x y\). 9. Simplify \(\sqrt[3]{256 x^{5}} - \sqrt[3]{4 x^{5}}\). Rewrite: $$256 = 2^{8}, \ 4 = 2^{2}.$$ So: $$\sqrt[3]{2^{8} x^{5}} - \sqrt[3]{2^{2} x^{5}} = 2^{8/3} x^{5/3} - 2^{2/3} x^{5/3} = x^{5/3} (2^{8/3} - 2^{2/3}) = x^{5/3} 2^{2/3} (2^{2} - 1).$$ Calculate \(2^{2} - 1 = 4 - 1 = 3\), so expression is $$3 \cdot 2^{2/3} x^{5/3} = 3 x^{5/3} \sqrt[3]{4}.$$ No options given, final simplified form is above. "q_count":9