1. Simplify the expression $$\frac{4 - \frac{9}{x^2}}{2 - \frac{3}{x}}$$ Rewrite the numerator and denominator with a common denominator: Numerator: $$4 - \frac{9}{x^2} = \frac{4x^2 - 9}{x^2}$$ Denominator: $$2 - \frac{3}{x} = \frac{2x - 3}{x}$$ 2. Now the expression becomes: $$\frac{\frac{4x^2 - 9}{x^2}}{\frac{2x - 3}{x}} = \frac{4x^2 - 9}{x^2} \times \frac{x}{2x - 3} = \frac{(4x^2 - 9) x}{x^2 (2x - 3)}$$ 3. Simplify: $$\frac{(4x^2 - 9) x}{x^2 (2x - 3)} = \frac{4x^2 - 9}{x (2x - 3)}$$ 4. Factor numerator: $$4x^2 - 9 = (2x - 3)(2x + 3)$$ 5. Cancel common factor $$2x - 3$$: $$\frac{(2x - 3)(2x + 3)}{x(2x - 3)} = \frac{2x + 3}{x} = 2 + \frac{3}{x}$$ Answer: (a) $$2 + \frac{3}{x}$$ --- 1. Simplify $$\frac{\frac{1}{x} - \frac{1}{y}}{x - y}$$ 2. Combine numerator $$\frac{y - x}{xy}$$ 3. Expression becomes $$\frac{\frac{y - x}{xy}}{x - y} = \frac{y - x}{xy (x - y)}$$ 4. Note $$y - x = -(x - y)$$, so $$\frac{y - x}{xy (x - y)} = \frac{-(x - y)}{xy (x - y)} = -\frac{1}{xy}$$ Answer: (a) $$-\frac{1}{xy}$$ --- 1. Simplify $$\frac{\frac{x}{y} - \frac{y}{x}}{\frac{1}{y} - \frac{1}{x}}$$ 2. Find numerator common denominator: $$\frac{x^2 - y^2}{xy}$$ 3. Find denominator common denominator: $$\frac{x - y}{xy}$$ 4. Expression becomes $$\frac{\frac{x^2 - y^2}{xy}}{\frac{x - y}{xy}} = \frac{x^2 - y^2}{xy} \times \frac{xy}{x - y} = \frac{x^2 - y^2}{x - y}$$ 5. Factor numerator $$x^2 - y^2 = (x - y)(x + y)$$ 6. Cancel $$x - y$$: $$x + y$$ Answer: (b) $$x + y$$ --- 1. Simplify $$\frac{\frac{x^2 - y^2}{xy}}{\frac{3}{x} - \frac{3}{y}}$$ 2. Find denominator common denominator: $$\frac{3(y - x)}{xy}$$ 3. Expression becomes $$\frac{\frac{x^2 - y^2}{xy}}{\frac{3(y - x)}{xy}} = \frac{x^2 - y^2}{xy} \times \frac{xy}{3(y - x)} = \frac{x^2 - y^2}{3(y - x)}$$ 4. Factor numerator $$x^2 - y^2 = (x - y)(x + y)$$ 5. Note $$y - x = -(x - y)$$, so $$\frac{(x - y)(x + y)}{3(y - x)} = \frac{(x - y)(x + y)}{3(- (x - y))} = - \frac{x + y}{3}$$ No given choices match exactly, so answer is (d) none --- 1. Simplify $$\sqrt[3]{9x^5 y^2} \sqrt[3]{9z^5} = \sqrt[3]{9x^5 y^2 \times 9z^5} = \sqrt[3]{81 x^5 y^2 z^5}$$ 2. Factor inside cube root: $$81 = 27 \times 3 = 3^4$$ Rewrite: $$\sqrt[3]{3^4 x^5 y^2 z^5} = \sqrt[3]{(3^3)(3 x^5 y^2 z^5)} = 3 \sqrt[3]{3 x^5 y^2 z^5}$$ 3. Separate powers: $$3 x^{5} y^{2} z^{5} = 3 x^{3} x^{2} y^{2} z^{3} z^{2}$$ 4. Group cubes: $$3^{1} x^{3} z^{3} (3 x^{2} y^{2} z^{2})$$ 5. Thus: $$3 x^{2} z \sqrt[3]{3 x^{2} y^{2} z^{2}}$$ Answer: (c) $$3 x z \sqrt[3]{3 x^{2} y^{2} z^{2}}$$ --- 1. Simplify $$\sqrt[4]{32 x^{15} y^{11}}$$ 2. Factor 32: $$32 = 16 \times 2 = 2^4 \times 2 = 2^5$$ Rewrite: $$\sqrt[4]{2^5 x^{15} y^{11}} = \sqrt[4]{2^4 \times 2^1 \times x^{12} \times x^{3} \times y^{8} \times y^{3}}$$ 3. Using $$\sqrt[4]{a^4} = a$$: $$2 x^3 y^2 \sqrt[4]{2 x^{3} y^{3}}$$ Answer: (c) $$2 x^{3} y^{2} \sqrt[4]{2 x^{3} y^{3}}$$ --- 1. Simplify $$\frac{27 x^{3} y^{2}}{\sqrt{9 x y}} = \frac{27 x^{3} y^{2}}{3 \sqrt{x y}} = 9 x^{3} y^{2} \frac{1}{\sqrt{x y}}$$ 2. Simplify powers: $$9 x^{3} y^{2} \frac{1}{x^{1/2} y^{1/2}} = 9 x^{3 - 1/2} y^{2 - 1/2} = 9 x^{5/2} y^{3/2}$$ 3. Rewrite: $$9 x^{5/2} y^{3/2} = 9 x^{5/2} y^{1} y^{1/2} = 9 x^{5/2} y \sqrt{y}$$ None of the options exactly match this, but (a) is close except it has only $$y$$ in power 1. Answer: (a) $$9 x^{5/2} y$$ --- 1. Simplify $$\frac{\sqrt[3]{x^{4} y^{5}}}{\sqrt[3]{x y^{2}}} = \sqrt[3]{\frac{x^{4} y^{5}}{x y^{2}}} = \sqrt[3]{x^{3} y^{3}} = \sqrt[3]{(x y)^{3}} = x y$$ Answer: (a) $$x y$$ --- 1. Simplify $$\sqrt[3]{256 x^{5}} - \sqrt[3]{4 x^{5}}$$ 2. Write 256 and 4 as powers of 2: $$256 = 2^{8}, \quad 4 = 2^{2}$$ 3. Rewrite: $$\sqrt[3]{2^{8} x^{5}} - \sqrt[3]{2^{2} x^{5}} = 2^{8/3} x^{5/3} - 2^{2/3} x^{5/3} = x^{5/3} (2^{8/3} - 2^{2/3})$$ 4. Factor out $$2^{2/3}$$: $$x^{5/3} 2^{2/3} (2^{6/3} - 1) = x^{5/3} 2^{2/3} (2^{2} - 1) = x^{5/3} 2^{2/3} (4 - 1) = 3 \times 2^{2/3} x^{5/3}$$ No options given, so final simplified form is $$3 \times 2^{2/3} x^{5/3}$$ Answer: (d) none