1. The problem asks which equation represents an inverse variation. Inverse variation means $y$ varies inversely as $x$, so $y = \frac{k}{x}$. Among the options, only b. $y = \frac{1}{x}$ fits this form. 2. Direct proportionality to $x^2$ means $y = kx^2$, which is a direct variation. Answer: a. Direct Variation. 3. Rain amount and water level in a dam increase together, so it's a direct variation. Answer: a. Direct Variation. 4. The constant $k$ in direct variation $y = kx$ is found by $k = \frac{y}{x}$. Answer: c. $k = \frac{y}{z}$ is incorrect; correct is $k = \frac{y}{x}$ but not listed exactly, closest is c. 5. Volume $V$ varies jointly as $r^2$ and $h$, so $V = kr^2h$. Answer: c. $V = kr^2h$. 6. Given $y$ varies directly as $x$, $y = kx$, and $y=15$ when $x=5$, so $k=3$. Find $y$ when $x=7$: $y=3 \times 7=21$. Answer: c. 21. 7. $y$ varies inversely as $x$, so $y = \frac{k}{x}$. Given $y=5$, $k=60$, find $x$: $x=\frac{k}{y}=\frac{60}{5}=12$. Answer: a. 12. 8. Weight $W$ varies directly with mass $M$: $W = kM$. Given $W=24$ when $M=4$, $k=6$. Find $M$ when $W=16$: $M=\frac{W}{k}=\frac{16}{6}=\frac{8}{3} \approx 2.67$ (not in options). Closest is a. 3 kg. 9. Hours $H$ inversely proportional to men $m$: $H = \frac{k}{m}$. Given $H=10$ when $m=2$, $k=20$. Find $m$ when $H=4$: $m=\frac{k}{H}=\frac{20}{4}=5$. Answer: c. 5. 10. Area $A$ varies jointly as base $b$ and height $h$: $A = kbh$. Given $A=12$ when $b=6$, $h=4$, so $k=\frac{12}{24}=0.5$. New base $b=8$, new height $h=8$, new area $A=0.5 \times 8 \times 8=32$. Answer: c. 32m². 11. Rule "add exponents of same base" is product of powers. Answer: c. Product of a power. 12. Simplify $x^{10}x^5 = x^{10+5} = x^{15}$. Answer: a. $x^{15}$. 13. Simplify $\frac{a^2 a^3 b^5 c^2 d^5}{a^4 c^5 b^2 d^3 e^2} = a^{2+3-4} b^{5-2} c^{2-5} d^{5-3} e^{-2} = a^{1} b^{3} c^{-3} d^{2} e^{-2} = \frac{a b^3 d^2}{c^3 e^2}$. Answer: b. $\frac{2a^2 b d^2 e^2}{c^2}$ is incorrect; correct is $\frac{a b^3 d^2}{c^3 e^2}$ closest is c. 14. Evaluate $|x^3|(1/x^3) = |x^3| \times x^{-3} = |x^3| x^{-3}$. If $x$ positive, $|x^3|=x^3$, so $x^3 x^{-3} = x^0=1$. None of the options match; likely a typo. 15. Evaluate $(a^3)^{1/5} = a^{3 \times \frac{1}{5}} = a^{3/5}$. Answer: a. $a^{3/5}$. 16. Simplify $(a^3 b^3 c^0)^{152} = a^{3 \times 152} b^{3 \times 152} c^0 = a^{456} b^{456}$. None of the options match; closest is a. 17. Equivalent radical of $2^2$ is $\sqrt{4}$. Answer: d. $\sqrt{4}$. 18. Equivalent radical of $(2x)^{1/3}$ is $\sqrt[3]{2x}$. Answer: c. $\sqrt[3]{10x^1}$ incorrect; correct is $\sqrt[3]{2x}$. 19. Transform $\sqrt[3]{27a^1}$ to rational exponent: $27^{1/3} a^{1/3} = 3 a^{1/3}$. Answer: a. $27 a^{1/3}$ incorrect; correct is $3 a^{1/3}$. 20. Equivalent to $\sqrt[3]{8 a^5} = \sqrt[3]{8} \sqrt[3]{a^5} = 2 a^{5/3}$. Answer: c. $2 a^3$ incorrect; correct is $2 a^{5/3}$. 21. Simplify $7 \sqrt{18} = 7 \times 3 \sqrt{2} = 21 \sqrt{2}$. Answer: b. $21 \sqrt{2}$. 22. Rational form of $\frac{1}{\sqrt{3}}$ is $\frac{\sqrt{3}}{3}$. Answer: b. $\frac{\sqrt{3}}{3}$. 23. Simplify $\sqrt{8 a^3} = \sqrt{8} \sqrt{a^3} = 2 \sqrt{2} a^{3/2} = 2 a \sqrt{2 a}$. Answer: b. $2 a \sqrt{2 a}$. 24. Simplify $\sqrt{16 a^5 b^3} = 4 a^2 b \sqrt{a b}$. Answer: a. $2 a b \sqrt{2 a^3}$ incorrect; correct is $4 a^2 b \sqrt{a b}$. 25. Simplify $\sqrt[75]{3} / \sqrt{3} = 3^{1/75} / 3^{1/2} = 3^{(1/75 - 1/2)} = 3^{-37/75}$. No matching options. 26. Evaluate $\frac{2 + \sqrt{2}}{\sqrt{5}}$ rationalizing denominator: Multiply numerator and denominator by $\sqrt{5}$: $\frac{(2 + \sqrt{2}) \sqrt{5}}{5} = \frac{2 \sqrt{5} + \sqrt{10}}{5}$. Answer: c. $\frac{2 \sqrt{5} + \sqrt{10}}{5}$. 27. Conjugate of $\frac{-2}{5} \sqrt{2}$ is $\frac{2}{5 + \sqrt{2}}$. Answer: c. 28. Simplify $\frac{2}{5 + \sqrt{2}}$ multiply numerator and denominator by $5 - \sqrt{2}$: $\frac{2(5 - \sqrt{2})}{25 - 2} = \frac{10 - 2 \sqrt{2}}{23}$. Answer: d. 29. Sum $2 \sqrt{3} + 4 \sqrt[3]{3^2}$ cannot be combined as radicals differ. Answer: none match; likely d. $5 \sqrt{3}$ if ignoring cube root. 30. Evaluate $3 \sqrt{2} + 5 \sqrt{3} + 2 \sqrt{2} - 3 \sqrt{3} = (3+2) \sqrt{2} + (5-3) \sqrt{3} = 5 \sqrt{2} + 2 \sqrt{3}$. Answer: a. 31. Evaluate $6 \sqrt{8} + 2 \sqrt{27} - 5 \sqrt{18}$: $6 \times 2 \sqrt{2} + 2 \times 3 \sqrt{3} - 5 \times 3 \sqrt{2} = 12 \sqrt{2} + 6 \sqrt{3} - 15 \sqrt{2} = -3 \sqrt{2} + 6 \sqrt{3}$. Answer: c. 32. True statement: Add radicals only with same radicand and same indices. Answer: b. 33. Solve $\sqrt{x} = 49^2 = 2401$. Square both sides: $x = 2401^2$ huge number, but options suggest $x=7$. Likely $\sqrt{x} = 49$, so $x=49^2=2401$. Answer: d. 7 incorrect; question ambiguous. 34. First step to solve $\sqrt{x + 2} = 6$ is square both sides. Answer: b. 35. After squaring: $x + 2 = 36$, so $x=34$. Answer: a. 36. Given $\sqrt{2x + 3} = \sqrt{x + 6}$, square both sides: $2x + 3 = x + 6$, so $x=3$. Answer: d. 37. Who traveled north? Plane. Answer: b. 38. Theorem to use: Pythagorean Theorem. Answer: b. 39. Equation to use: $\sqrt{8^2 + 6^2} = d$. Answer: c. 40. Distance $d = \sqrt{8^2 + 6^2} = \sqrt{64 + 36} = \sqrt{100} = 10$ miles. Answer: d.