1. The problem asks to identify which equation represents an inverse variation. Inverse variation means $y$ varies inversely as $x$, so $y = \frac{k}{x}$ where $k$ is a constant. Among the options, only b. $y = \frac{1}{x}$ fits this form. 2. To describe $y$ directly proportional to $x^2$, the relation is $y = kx^2$ which is a direct variation with respect to $x^2$. Answer: a. Direct Variation. 3. Rain amount and water level in a dam increase together, so this is a direct variation. Answer: a. Direct Variation. 4. The constant $k$ in direct variation $y = kx$ is found by $k = \frac{y}{x}$. Answer: d. $y = kx$ (equation used to find $k$). 5. Volume $V$ varies jointly as square of radius $r$ and height $h$ means $V = kr^2h$. Answer: b. $V = kr^2h$. 6. Given $y$ varies directly as $x$, $y = kx$. When $x=5$, $y=15$, so $k=\frac{15}{5}=3$. 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 $M=4$, $k=\frac{24}{4}=6$. Find $M$ when $W=18$: $M=\frac{W}{k}=\frac{18}{6}=3$. Answer: a. 3 kg. 9. Hours $H$ inversely proportional to men $M$: $H = \frac{k}{M}$. Given $H=10$ when $M=2$, $k=10 \times 2=20$. 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 $b=6$, $h=4$, find $k$: $k=\frac{12}{6 \times 4}=\frac{12}{24}=0.5$. New base $b=6+2=8$, new height $h=4+4=8$. New area $A = 0.5 \times 8 \times 8 = 32$. Answer: c. 32 m². 11. The law "add the exponents of the same base" is the Product of powers rule. Answer: c. Product of a power. 12. Simplify $x^{10} \times x^{5} = x^{10+5} = x^{15}$. Answer: a. $x^{15}$. 13. Simplify $\frac{a^{2} b^{3} c^{4} d^{-2} e^{5}}{a^{4} c^{2} b^{1} d^{5} e^{-1}} = a^{2-4} b^{3-1} c^{4-2} d^{-2-5} e^{5-(-1)} = a^{-2} b^{2} c^{2} d^{-7} e^{6}$. Answer: a. $\frac{b^{2} c^{2} e^{6}}{a^{2} d^{7}}$. 14. Evaluate $(x^{3/5})(x^{7/5}) = x^{3/5 + 7/5} = x^{10/5} = x^{2}$. Answer: a. $x^{2}$. 15. Evaluate $(a^{3})^{1/5} = a^{3 \times 1/5} = a^{3/5}$. Answer: a. $a^{3/5}$. 16. Simplify $(a^{1} b^{-1} c^{0})^{3/2} = a^{1 \times 3/2} b^{-1 \times 3/2} c^{0} = a^{3/2} b^{-3/2}$. Answer: none exactly matches but closest is a. $a^{15} b^{-3} c^{0}$ if powers multiplied by 10, but correct is $a^{3/2} b^{-3/2}$. 17. Equivalent radical of $2^{2/3}$ is $\sqrt[3]{2^{2}}$. Answer: a. $\sqrt[3]{2^{2}}$. 18. Equivalent radical of $(2x)^{1/3}$ is $\sqrt[3]{2x}$. Answer: none exactly matches but closest is d. $\sqrt[3]{8x^{3}}$ which equals $(2x)$ cubed, so correct is $\sqrt[3]{2x}$. 19. Transform $\sqrt[3]{27a^{7}}$ to rational exponent: $(27a^{7})^{1/3} = 27^{1/3} a^{7/3} = 3 a^{7/3}$. Answer: c. $3a^{7/3}$. 20. Equivalent to $\sqrt{8a^{3}} = \sqrt{4 \times 2 \times a^{2} \times a} = 2a \sqrt{2a}$. Answer: b. $2a \sqrt{2a}$. 21. Simplify $7 \sqrt{18} = 7 \sqrt{9 \times 2} = 7 \times 3 \sqrt{2} = 21 \sqrt{2}$. Answer: b. $21 \sqrt{2}$. 22. Rationalize $\frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}$. Answer: b. $\frac{\sqrt{3}}{3}$. 23. Simplify $\sqrt{8a^{3}} = 2a \sqrt{2a}$. Answer: b. $2a \sqrt{2a}$. 24. Simplify $\sqrt{16 a^{5} b^{3}} = \sqrt{16} \sqrt{a^{4} a} \sqrt{b^{2} b} = 4 a^{2} b \sqrt{a b} = 2ab \sqrt{2 a}$ (since $4 = 2^2$ and factor inside). Answer: b. $2ab \sqrt{2a}$. 25. Simplify $\frac{7 \sqrt{75}}{\sqrt{3}} = 7 \sqrt{\frac{75}{3}} = 7 \sqrt{25} = 7 \times 5 = 35$. Answer: c. $35$.