1. Problem 39: Simplify $$\frac{4 - \frac{9}{x^2}}{2 - \frac{3}{x}}$$ and identify the correct expression from the options. Step 1: Write the numerator and denominator with common denominators: $$4 - \frac{9}{x^2} = \frac{4x^2 - 9}{x^2},\quad 2 - \frac{3}{x} = \frac{2x - 3}{x}$$ Step 2: Rewrite the original expression: $$\frac{\frac{4x^2 - 9}{x^2}}{\frac{2x - 3}{x}} = \frac{4x^2 - 9}{x^2} \times \frac{x}{2x - 3} = \frac{4x^2 - 9}{x} \times \frac{1}{2x - 3}$$ Step 3: Factor numerator $4x^2 - 9 = (2x - 3)(2x + 3)$: $$\frac{(2x - 3)(2x + 3)}{x(2x - 3)}$$ Step 4: Cancel common factor $(2x - 3)$: $$\frac{2x + 3}{x} = 2 + \frac{3}{x}$$ Step 5: Match this with options: (a) $2 + \frac{3}{x}$ is correct. --- 2. Problem 40: Simplify $$\frac{\frac{1}{x} - \frac{1}{y}}{x - y}$$ Step 1: Combine numerator: $$\frac{y - x}{xy}$$ Step 2: Expression becomes $$\frac{\frac{y - x}{xy}}{x - y} = \frac{y - x}{xy(x - y)}$$ Step 3: Note that $y - x = -(x - y)$, so: $$\frac{-(x - y)}{xy(x - y)} = -\frac{1}{xy}$$ Step 4: Match with options: (a) $-\frac{1}{xy}$ is correct. --- 3. Problem 41: Simplify $$\frac{\frac{x}{y} - \frac{y}{x}}{\frac{1}{y} - \frac{1}{x}}$$ Step 1: Find numerator: $$\frac{x^2 - y^2}{xy}$$ Step 2: Find denominator: $$\frac{x - y}{xy}$$ Step 3: Division becomes: $$\frac{\frac{x^2 - y^2}{xy}}{\frac{x - y}{xy}} = \frac{x^2 - y^2}{x - y} = \frac{(x - y)(x + y)}{x - y} = x + y$$ Step 4: Match result with options: (b) $x + y$ is correct. --- 4. Problem 42: Simplify $$\frac{x^2 - y^2}{xy} \div \left( \frac{3}{x} - \frac{3}{y} \right)$$ Step 1: Factor numerator: $$\frac{(x - y)(x + y)}{xy}$$ Step 2: Simplify denominator: $$\frac{3(y - x)}{xy} = -\frac{3(x - y)}{xy}$$ Step 3: Division becomes multiplication by reciprocal: $$\frac{(x - y)(x + y)}{xy} \times \frac{xy}{-3(x - y)} = \frac{(x - y)(x + y)}{xy} \times \frac{xy}{-3(x - y)}$$ Step 4: Cancel $xy$ and $(x - y)$: $$\frac{x + y}{-3} = -\frac{x + y}{3}$$ Step 5: None of the options match $ -\frac{x + y}{3}$, so (d) none is correct. --- 5. Problem 43: Simplify $$\sqrt[3]{9x^5y^2} \times \sqrt[3]{9z^5}$$ Step 1: Multiply inside cube roots: $$\sqrt[3]{9x^5y^2 \times 9z^5} = \sqrt[3]{81x^5y^2z^5}$$ Step 2: Factor 81 as $3^4$ and rewrite powers: $$\sqrt[3]{3^4 x^5 y^2 z^5} = 3 \sqrt[3]{3 x^5 y^2 z^5}$$ Step 3: Split powers inside root as cubes and remainders: $$3 \sqrt[3]{3 x^5 y^2 z^5} = 3 x^{1+1+ ...} z^{1+1+ ...} \sqrt[3]{3 y^2 x^{2} z^{2}}$$ Actually, write correctly: powers divisible by 3: $$x^5 = x^{3+2} = x^{3} x^{2} \Rightarrow x^{1} \text{ outside root}, x^{2} \text{ inside} $$ $$z^5 = z^{3+2} = z^{3} z^{2}$$ Step 4: So: $$3 x z \sqrt[3]{3 x^{2} y^{2} z^{2}}$$ Step 5: Match with options: (c) $3 x z \sqrt[3]{3 x^{2} y^{2} z^{2}}$ is correct. --- 6. Problem 44: Simplify $$\sqrt[4]{32 x^{15} y^{11}}$$ Step 1: Express powers in terms of multiples of 4: $$32 = 2^5$$ $$x^{15} = x^{12} x^3 = (x^4)^3 x^3$$ $$y^{11} = y^{8} y^{3} = (y^4)^2 y^3$$ Step 2: Take fourth root: $$\sqrt[4]{2^{5} x^{15} y^{11}} = 2^{1} x^{3} y^{2} \sqrt[4]{2 x^{3} y^{3}} = 2 x^{3} y^{2} \sqrt[4]{2 x^{3} y^{3}}$$ Step 3: Match with options: (c) $2 x^{3} y^{2} \sqrt[4]{2 x^{3} y^{3}}$ is correct. --- 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: Expression becomes: $$\frac{27 x^3 y^2}{3 \sqrt{x y}} = 9 x^3 y^2 \div \sqrt{x y} = 9 x^3 y^2 \times \frac{1}{\sqrt{x y}}$$ Step 3: Simplify powers: $$x^{3} \div x^{1/2} = x^{5/2}, \quad y^{2} \div y^{1/2} = y^{3/2} = y \sqrt{y}$$ Step 4: So expression is: $$9 x^{5/2} y^{3/2} = 9 x^{5/2} y \sqrt{y}$$ Step 5: Options show similar terms; closest is (b) $9 x^{2} y \sqrt{x y}$ but powers differ. Step 6: Re-express carefully: $$y^{3/2} = y \sqrt{y}, \quad \text{our expression is } 9 x^{5/2} y \sqrt{y}$$ No option matches exactly; next check is (a) $9 x^{5/2} y$ (missing $\sqrt{y}$), (b) $9 x^{2} y \sqrt{x y}$ (missing $x^{1/2}$ factor and differs). Step 7: Since none matches perfectly, choose (a) $9 x^{5/2} y$ assuming typo or omission is likely. --- 8. Problem 46: Simplify $$\frac{\sqrt[3]{x^4 y^5}}{\sqrt[3]{x y^2}}$$ Step 1: Write as one cube root: $$\sqrt[3]{\frac{x^4 y^5}{x y^2}} = \sqrt[3]{x^{3} y^{3}} = \sqrt[3]{(x y)^3}$$ Step 2: Simplify cube root: $$x y$$ Step 3: Match with options: (a) $x y$ is correct. --- 9. Problem 47: Simplify $$\sqrt[3]{256 x^5} - \sqrt[3]{4 x^5}$$ Step 1: Rewrite cube roots: $$\sqrt[3]{256 x^5} = \sqrt[3]{2^{8} x^{5}} = 2^{8/3} x^{5/3} = 2^{2 + 2/3} x^{1 + 2/3} = 4 \times 2^{2/3} x \times x^{2/3} = 4 x \sqrt[3]{4 x^{2}}$$ Step 2: Similarly, $$\sqrt[3]{4 x^5} = \sqrt[3]{2^{2} x^{5}} = 2^{2/3} x^{5/3} = x \times 2^{2/3} x^{2/3} = x \sqrt[3]{4 x^{2}}$$ Step 3: Expression is: $$4 x \sqrt[3]{4 x^{2}} - x \sqrt[3]{4 x^{2}} = (4x - x) \sqrt[3]{4 x^{2}} = 3 x \sqrt[3]{4 x^{2}}$$ Final Answers: 39: (a) 40: (a) 41: (b) 42: (d) 43: (c) 44: (c) 45: (a) 46: (a) 47: $3 x \sqrt[3]{4 x^{2}}$ (not explicitly an option)