Radical Operations

1. Simplify $\frac{\sqrt{24}}{\sqrt{12}}$. Step 1: Use the property $\frac{\sqrt{a}}{\sqrt{b}}=\sqrt{\frac{a}{b}}$. Step 2: Calculate $\sqrt{\frac{24}{12}} = \sqrt{2}$. Answer: $\sqrt{2}$. 2. Simplify $\frac{\sqrt{20}}{\sqrt{5}}$. Step 1: Use $\frac{\sqrt{a}}{\sqrt{b}}=\sqrt{\frac{a}{b}}$. Step 2: Calculate $\sqrt{\frac{20}{5}}=\sqrt{4}=2$. Answer: $2$. 3. Simplify $\frac{4\sqrt{75}}{2\sqrt{3}}$. Step 1: Simplify inside radical: $\sqrt{75} = \sqrt{25\times3} = 5\sqrt{3}$. Step 2: Substitute: $\frac{4\times5\sqrt{3}}{2\sqrt{3}}= \frac{20\sqrt{3}}{2\sqrt{3}}$. Step 3: Cancel $\sqrt{3}$ and simplify coefficients: $\frac{20}{2}=10$. Answer: $10$. 4. Simplify $\frac{\sqrt{16a^3}}{\sqrt{4a}}$. Step 1: Write as $\sqrt{\frac{16a^3}{4a}}$. Step 2: Simplify inside the root: $\frac{16a^3}{4a} = 4a^2$. Step 3: Take the root: $\sqrt{4a^2} = 2a$. Answer: $2a$. 5. Simplify $\frac{\sqrt{4b^3 c}}{\sqrt{2bc}}$. Step 1: Write as $\sqrt{\frac{4b^3 c}{2bc}}$. Step 2: Simplify inside: $\frac{4b^3 c}{2bc} = 2b^2$. Step 3: Root: $\sqrt{2b^2} = b\sqrt{2}$. Answer: $b\sqrt{2}$. 6. Simplify $\frac{8\sqrt[3]{16}}{2\sqrt[3]{2}}$. Step 1: Use $\frac{8}{2} = 4$. Step 2: Simplify cube roots: $\frac{\sqrt[3]{16}}{\sqrt[3]{2}} = \sqrt[3]{\frac{16}{2}} = \sqrt[3]{8} = 2$. Step 3: Multiply: $4 \times 2 = 8$. Answer: $8$. 7. Simplify $\frac{2}{\sqrt{3}}$. Step 1: Rationalize denominator by multiplying numerator and denominator by $\sqrt{3}$. Step 2: $\frac{2}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{2\sqrt{3}}{3}$. Answer: $\frac{2\sqrt{3}}{3}$. 8. Simplify $\frac{4}{\sqrt{2}}$. Step 1: Rationalize denominator by multiplying numerator and denominator by $\sqrt{2}$. Step 2: $\frac{4}{\sqrt{2}} \times \frac{\sqrt{2}}{\sqrt{2}} = \frac{4\sqrt{2}}{2} = 2\sqrt{2}$. Answer: $2\sqrt{2}$. 9. Simplify $\frac{\sqrt{3}}{\sqrt{5}}$. Step 1: Write as $\sqrt{\frac{3}{5}}$. Answer: $\sqrt{\frac{3}{5}}$. 10. Simplify $\frac{2}{\sqrt{3} + 5}$. Step 1: Rationalize denominator by multiplying numerator and denominator by conjugate $\sqrt{3} - 5$. Step 2: Numerator: $2(\sqrt{3} - 5) = 2\sqrt{3} - 10$. Step 3: Denominator: $(\sqrt{3} + 5)(\sqrt{3} - 5) = 3 - 25 = -22$. Step 4: Final: $\frac{2\sqrt{3} - 10}{-22} = \frac{10 - 2\sqrt{3}}{22} = \frac{5 - \sqrt{3}}{11}$. Answer: $\frac{5 - \sqrt{3}}{11}$. 11. Simplify $\frac{3}{2 - \sqrt{5}}$. Step 1: Multiply numerator and denominator by conjugate $2 + \sqrt{5}$. Step 2: Numerator: $3(2 + \sqrt{5}) = 6 + 3\sqrt{5}$. Step 3: Denominator: $(2 - \sqrt{5})(2 + \sqrt{5}) = 4 - 5 = -1$. Step 4: Final: $\frac{6 + 3\sqrt{5}}{-1} = -6 - 3\sqrt{5}$. Answer: $-6 - 3\sqrt{5}$. 12. Simplify $\frac{\sqrt{3}}{\sqrt{3} - \sqrt{5}}$. Step 1: Multiply numerator and denominator by conjugate $\sqrt{3} + \sqrt{5}$. Step 2: Numerator: $\sqrt{3}(\sqrt{3} + \sqrt{5}) = 3 + \sqrt{15}$. Step 3: Denominator: $(\sqrt{3} - \sqrt{5})(\sqrt{3} + \sqrt{5}) = 3 - 5 = -2$. Step 4: Final: $\frac{3 + \sqrt{15}}{-2} = -\frac{3 + \sqrt{15}}{2}$. Answer: $-\frac{3 + \sqrt{15}}{2}$. 13. Simplify $\frac{7 + \sqrt{2}}{3\sqrt{3} - \sqrt{2}}$. Step 1: Multiply numerator and denominator by conjugate $3\sqrt{3} + \sqrt{2}$. Step 2: Numerator: $(7 + \sqrt{2})(3\sqrt{3} + \sqrt{2})$. Expand: $7 \times 3\sqrt{3} = 21\sqrt{3}$, $7 \times \sqrt{2} = 7\sqrt{2}$, $\sqrt{2} \times 3\sqrt{3} = 3\sqrt{6}$, $\sqrt{2} \times \sqrt{2} = 2$. Total: $21\sqrt{3} + 7\sqrt{2} + 3\sqrt{6} + 2$. Step 3: Denominator: $(3\sqrt{3})^2 - (\sqrt{2})^2 = 9 \times 3 - 2 = 27 - 2 = 25$. Step 4: Final: $\frac{21\sqrt{3} + 7\sqrt{2} + 3\sqrt{6} + 2}{25}$. Answer: $\frac{21\sqrt{3} + 7\sqrt{2} + 3\sqrt{6} + 2}{25}$. 14. Simplify $\frac{\sqrt{x} + 1}{\sqrt{x} - 1}$. Step 1: Multiply numerator and denominator by conjugate $\sqrt{x} + 1$. Step 2: Numerator: $(\sqrt{x} + 1)^2 = x + 2\sqrt{x} + 1$. Step 3: Denominator: $(\sqrt{x} - 1)(\sqrt{x} + 1) = x - 1$. Answer: $\frac{x + 2\sqrt{x} + 1}{x - 1}$. 15. Simplify $\frac{2\sqrt{x}}{x - \sqrt{5}}$. Step 1: Multiply numerator and denominator by conjugate $x + \sqrt{5}$. Step 2: Numerator: $2\sqrt{x}(x + \sqrt{5}) = 2x\sqrt{x} + 2\sqrt{5x}$. Step 3: Denominator: $(x - \sqrt{5})(x + \sqrt{5}) = x^2 - 5$. Answer: $\frac{2x\sqrt{x} + 2\sqrt{5x}}{x^2 - 5}$.