1. Simplify: $ (3\sqrt{2} – 4\sqrt{3}) (2\sqrt{2} + 5) $. Multiply each term: $$= 3\sqrt{2} \times 2\sqrt{2} + 3\sqrt{2} \times 5 - 4\sqrt{3} \times 2\sqrt{2} -4\sqrt{3} \times 5$$ Simplify each: $3 \times 2 \times (\sqrt{2} \times \sqrt{2}) = 6 \times 2 = 12$ $3\sqrt{2} \times 5 = 15\sqrt{2}$ $-4 \times 2 \times \sqrt{3} \times \sqrt{2} = -8\sqrt{6}$ $-4\sqrt{3} \times 5 = -20\sqrt{3}$ Combine: $$12 + 15\sqrt{2} - 8\sqrt{6} - 20\sqrt{3}$$ Answer is C. 2. Rationalize $\frac{25}{\sqrt{7}}$. Multiply numerator and denominator by $\sqrt{7}$: $$\frac{25}{\sqrt{7}} \times \frac{\sqrt{7}}{\sqrt{7}} = \frac{25\sqrt{7}}{7}$$ Answer is C. 3. Multiply $\sqrt{50} \times 5\sqrt{6}$. $\sqrt{50} = 5\sqrt{2}$ Product: $$5\sqrt{2} \times 5\sqrt{6} = 25\sqrt{12}$$ Answer is B. 4. Compute $(\sqrt{27} - \sqrt{8})(\sqrt{3} - \sqrt{2})$. Expand: $$\sqrt{27}\sqrt{3} - \sqrt{27}\sqrt{2} - \sqrt{8}\sqrt{3} + \sqrt{8}\sqrt{2}$$ Simplify: $\sqrt{27\times3} = \sqrt{81} = 9$ $\sqrt{27\times2} = \sqrt{54} = 3\sqrt{6}$ $\sqrt{8\times3} = \sqrt{24} = 2\sqrt{6}$ $\sqrt{8\times2} = \sqrt{16} = 4$ Combine: $$9 - 3\sqrt{6} - 2\sqrt{6} + 4 = (9+4) - (3\sqrt{6} + 2\sqrt{6}) = 13 - 5\sqrt{6}$$ Answer is A. 5. Simplify $-\sqrt{81} \cdot \sqrt{32} \cdot \sqrt{112}$. Product inside radical: $$81 \times 32 \times 112$$ Calculate: $81 \times 32 = 2592$ $2592 \times 112 = 290304$ Simplify radicals: $\sqrt{81} = 9$ So original expression: $$-9 \times \sqrt{32} \times \sqrt{112}$$ Combine under one radical: $$-9 \sqrt{32 \times 112} = -9 \sqrt{3584}$$ Factor: $$3584 = 256 \times 14$$ So: $$-9 \times 16 \sqrt{14} = -144 \sqrt{14}$$ Answer is B. 6. Simplify $\sqrt{6x} + \sqrt{24x} - \sqrt{24x}$. $\sqrt{24x} - \sqrt{24x} = 0$ So answer is $\sqrt{6x}$. Answer is C. 7. Simplify $\sqrt[3]{135x^3}$. Prime factor 135: $135 = 27 \times 5 = 3^3 \times 5$ So: $$\sqrt[3]{3^3 \times 5 \times x^3} = x \times 3 \times \sqrt[3]{5} = 3x \sqrt[3]{5}$$ Answer is B. 8. Solve $-5\sqrt{7} + ? = -\sqrt{7}$. Add $5\sqrt{7}$ both sides: $? = -\sqrt{7} + 5\sqrt{7} = 4\sqrt{7}$ Answer is C. 9. Radicals differ by $18\sqrt{y}$. Pairs: - $-\sqrt{9y}$ and $-\sqrt{9y}$ difference 0 - $\sqrt{y}$ and $-17\sqrt{y}$ difference $18\sqrt{y}$ Answer is B. 10. Solve $-2 \sqrt[3]{x} + 5 = -3$. Subtract 5: $-2 \sqrt[3]{x} = -8$ Divide: $\sqrt[3]{x} = 4$ Cube both sides: $$x = 4^3 = 64$$ Answer is D. 11. Solve $\sqrt{4y} + 9 = 11$. Subtract 9: $\sqrt{4y} = 2$ Square both sides: $4y = 4$ $y = 1$ None of options matches, closest logical is 26, but none correct from options. 12. A: $\sqrt{45x^2y^3} = 3xy\sqrt{5y}$ is true. B: $(\frac{2}{5})^{5/6} = \sqrt{(\frac{2}{5})^{5}}$ is false because $^{5/6}$ is sixth root, not square root. Answer A. 13. A: $(-29)^{1/3} = \sqrt[3]{-29}$ true. B: $(-29)^{1/3} = \sqrt[3]{29}$ false (sign matters). Answer A. 14. A: $\sqrt{16x^2y^2} = 4\sqrt{x^2y^2}$ false, because $\sqrt{16x^2y^2} = 4xy$ B: $\sqrt[4]{16x^2y^2} = 4\sqrt[4]{x^2y^2}$ false. Answer D. 15. A: $\sqrt{\frac{81x^8}{625y^{16}}} = \frac{3x^4}{5y^4}$ true. B: $\sqrt{100x^4y^{12}} = 10x^2y^6$ true. Answer C. 16. A: $(3x + 1)^{5/4} = \sqrt[4]{(3x +1)^5}$ true. B: $7x^{1/2}y^{1/3} = 7\sqrt[6]{x^3y^2}$ false. Answer A. 17. Simplify $\sqrt{\frac{4a^3b}{9a^4}} = \sqrt{\frac{4b}{9a}} = \frac{2}{3} \sqrt{\frac{b}{a}} = \frac{2\sqrt{ab}}{3a}$ Answer A. 18. Divide $\frac{3\sqrt{-5}}{\sqrt{7}+x}$: Expression is undefined as no simplification matches given options exactly, match closest is D. Answer D. 19. Solve $\sqrt[3]{2x} -5 = 5$. Add 5: $\sqrt[3]{2x} = 10$ Cube: $2x = 1000$ $x = 500$ Not an option. Answer D (none of preceding). 20. $\sqrt[3]{125} = (125)^{1/3}$ Closest is not given exactly; option D is $(125)^{1/7}$ incorrect. Answer D. 21. Index in $22^{(12)}\sqrt{17^2}$ is 12. Answer B. 22. Positive correlation implies direct variation. Answer D. 23. One increases, other decreases proportionally means indirect variation. Answer D. 24. Variation where one quantity varies directly as product of two quantities is joint variation. Answer C. 25. Simplify $\sqrt[4]{(-6x^2y^4)^4}$ $= -6x^2y^4$ Answer C. 26. Fourth root of $81m^{12}$ $= \sqrt[4]{3^4 m^{12}} = 3m^3$ Answer B. 27. Simplify $\sqrt{\frac{64x^3 y}{2x^4}} = \sqrt{32 \frac{y}{x}} = 4 \sqrt{\frac{2xy}{x^2}} = 4 \sqrt{\frac{2y}{x}}$ Best match B. 28. If 21 oranges cost 126, 1 orange costs 6. Php 30 buys $\frac{30}{6} = 5$ oranges. Answer C. 29. $ y = k x z , 60 = k \times 3 \times 4, 60 = 12k, k=5$ Answer B. 30. Simplify $(3 + \sqrt{2})(3 - \sqrt{2}) = 9 - 2 = 7$ not in options. Closest context answer is 4 (incorrect), so answer none, but closest B is 5. Answer B.