1. Problem 39: Simplify the expression \[\frac{4 - \frac{9}{x^2}}{2 - \frac{3}{x}}\]. Step 1: Find a common denominator in numerator and denominator separately. Numerator: $$4 - \frac{9}{x^2} = \frac{4x^2 - 9}{x^2}$$ Denominator: $$2 - \frac{3}{x} = \frac{2x - 3}{x}$$ Step 2: Rewrite the whole fraction: $$\frac{\frac{4x^2 - 9}{x^2}}{\frac{2x - 3}{x}} = \frac{4x^2 - 9}{x^2} \times \frac{x}{2x - 3} = \frac{4x^2 - 9}{x^2} \times \frac{x}{2x - 3}$$ Step 3: Simplify the multiplication: $$\frac{4x^2 - 9}{x^2} \times \frac{x}{2x - 3} = \frac{(4x^2 - 9) x}{x^2 (2x - 3)} = \frac{(4x^2 - 9)}{x (2x - 3)}$$ Step 4: Factor numerator: $$4x^2 - 9 = (2x - 3)(2x + 3)$$ So $$\frac{(2x - 3)(2x + 3)}{x (2x - 3)}$$ Step 5: Cancel common factor \(2x - 3\): $$\frac{2x + 3}{x} = 2 + \frac{3}{x}$$ Answer to 39: (a) \(2 + \frac{3}{x}\) 2. Problem 40: Simplify \[\frac{\frac{1}{x} - \frac{1}{y}}{x - y}\]. Step 1: Combine numerator: $$\frac{1}{x} - \frac{1}{y} = \frac{y - x}{xy}$$ Step 2: Write entire fraction: $$\frac{\frac{y - x}{xy}}{x - y} = \frac{y - x}{xy} \times \frac{1}{x - y}$$ Note \(y - x = -(x - y)\), so: $$= \frac{-(x - y)}{xy} \times \frac{1}{x - y} = -\frac{1}{xy}$$ Answer to 40: (a) \(-\frac{1}{xy}\) 3. Problem 41: Simplify \[\frac{\frac{x}{y} - \frac{y}{x}}{\frac{1}{y} - \frac{1}{x}}\]. Step 1: Simplify numerator: $$\frac{x}{y} - \frac{y}{x} = \frac{x^2 - y^2}{xy} = \frac{(x - y)(x + y)}{xy}$$ Step 2: Simplify denominator: $$\frac{1}{y} - \frac{1}{x} = \frac{x - y}{xy}$$ Step 3: Entire fraction: $$\frac{\frac{(x - y)(x + y)}{xy}}{\frac{x - y}{xy}} = \frac{(x - y)(x + y)}{xy} \times \frac{xy}{x - y}$$ Simplify: $$= x + y$$ Answer to 41: (b) \(x + y\) 4. Problem 42: Simplify \[\frac{\frac{x^2 - y^2}{xy}}{\frac{3}{x} - \frac{3}{y}}\]. Step 1: Numerator: $$\frac{x^2 - y^2}{xy} = \frac{(x - y)(x + y)}{xy}$$ Step 2: Denominator: $$\frac{3}{x} - \frac{3}{y} = 3\left(\frac{1}{x} - \frac{1}{y}\right) = 3 \frac{y - x}{xy} = \frac{3(y - x)}{xy}$$ Step 3: Entire fraction: $$\frac{\frac{(x - y)(x + y)}{xy}}{\frac{3(y - x)}{xy}} = \frac{(x - y)(x + y)}{xy} \times \frac{xy}{3(y - x)}$$ Recall \(y - x = -(x - y)\), so denominator term is \(3 (y - x) = -3(x - y)\): $$= \frac{(x - y)(x + y)}{xy} \times \frac{xy}{-3(x - y)} = -\frac{(x + y)}{3}$$ Since answer choices do not have negative sign explicitly and only (c) matches magnitude, Answer is: (d) none 5. Problem 43: Simplify $$\sqrt[3]{9x^5y^2} \times \sqrt[3]{9z^5} = \sqrt[3]{9 \times 9 \times x^5 \times y^2 \times z^5} = \sqrt[3]{81 x^5 y^2 z^5}$$ Factor 81 as $$3^4$$: $$= \sqrt[3]{3^4 x^5 y^2 z^5} = 3^{4/3} x^{5/3} y^{2/3} z^{5/3}$$ Rewrite: $$= 3 \times 3^{1/3} \times x^{1 + 2/3} \times y^{2/3} \times z^{1 + 2/3} = 3 x^{2} z \times \sqrt[3]{3 x^{2} y^{2} z^{2}}$$ Answer to 43: (c) \(3 x^{2} z \times \sqrt[3]{3 x^{2} y^{2} z^{2}}\) 6. Problem 44: Simplify \(\sqrt[4]{32 x^{15} y^{11}}\) Step 1: Express 32 as \(2^5\). Step 2: Distribute root: $$ = \sqrt[4]{2^5} \times \sqrt[4]{x^{15}} \times \sqrt[4]{y^{11}} = 2^{5/4} x^{15/4} y^{11/4}$$ Step 3: Break powers into integer and fractional parts: $$2^{5/4} = 2^{1} \times 2^{1/4} = 2 \times \sqrt[4]{2}$$ $$x^{15/4} = x^{3 + 3/4} = x^{3} \times x^{3/4}$$ $$y^{11/4} = y^{2 + 3/4} = y^{2} \times y^{3/4}$$ Step 4: Combine: $$= 2 x^{3} y^{2} \times \sqrt[4]{2 x^{3} y^{3}}$$ Answer to 44: (c) \(2 x^{3} y^{2} \times \sqrt[4]{2 x^{3} y^{3}}\) 7. Problem 45: Simplify $$\frac{27 x^{3} y^{2}}{\sqrt{9 x y}}$$ Step 1: Simplify the denominator: $$\sqrt{9 x y} = 3 \sqrt{x y}$$ Step 2: Rewrite expression: $$\frac{27 x^{3} y^{2}}{3 \sqrt{x y}} = 9 x^{3} y^{2} \times \frac{1}{\sqrt{x y}}$$ Step 3: Express radical as fractional power: $$= 9 x^{3} y^{2} \times (x y)^{-\frac{1}{2}} = 9 x^{3 - \frac{1}{2}} y^{2 - \frac{1}{2}} = 9 x^{\frac{5}{2}} y^{\frac{3}{2}}$$ Step 4: Write answer with radical: $$9 x^{\frac{5}{2}} y^{\frac{3}{2}} = 9 x^{2} y \times x^{\frac{1}{2}} y^{\frac{1}{2}} = 9 x^{2} y \times \sqrt{x y}$$ Answer to 45: (b) \(9 x^{2} y \times \sqrt{x y}\) 8. Problem 46: Simplify $$\frac{\sqrt[3]{x^{4} y^{5}}}{\sqrt[3]{x y^{2}}} = \sqrt[3]{\frac{x^{4} y^{5}}{x y^{2}}} = \sqrt[3]{x^{3} y^{3}}$$ Simplify inside cube root: $$= \sqrt[3]{(x y)^{3}} = x y$$ Answer to 46: (a) \(x y\) 9. Problem 47: Simplify $$\sqrt[3]{256 x^{5}} - \sqrt[3]{4 x^{5}}$$ Step 1: Express 256 and 4 as powers: $$256 = 2^{8}, 4 = 2^{2}$$ Step 2: Write cube roots: $$\sqrt[3]{2^{8} x^{5}} - \sqrt[3]{2^{2} x^{5}} = 2^{8/3} x^{5/3} - 2^{2/3} x^{5/3} = x^{5/3} (2^{8/3} - 2^{2/3})$$ Step 3: Factor out \(2^{2/3}\): $$= x^{5/3} 2^{2/3} (2^{6/3} - 1) = x^{5/3} 2^{2/3} (2^{2} - 1) = x^{5/3} 2^{2/3} (4 - 1) = 3 x^{5/3} 2^{2/3}$$ Step 4: Simplify: $$= 3 x^{5/3} \times \sqrt[3]{4}$$ Answer: \(3 x^{\frac{5}{3}} \times \sqrt[3]{4}\) (no options given to select) Final summary: "39": (a), "40": (a), "41": (b), "42": (d), "43": (c), "44": (c), "45": (b), "46": (a), "47": answered with simplified form