1. The problem asks to identify which equations represent inverse variation. Inverse variation means $y$ varies inversely as $x$, so $y = \frac{k}{x^n}$ for some $n > 0$. Among the options, only b. $y = \frac{k}{x}$ and d. $y = \frac{k}{x^2}$ are inverse variations. 2. Direct proportionality to $x^4$ means $y = kx^4$, which is a direct variation. Answer: a. Direct Variation. 3. Rain amount and water level in a dam increase together, so this is direct variation. Answer: a. Direct Variation. 4. The constant $k$ in direct variation $y = kx$ is found by $k = \frac{y}{x}$. Answer: c. $k = \frac{x}{y}$ is incorrect; correct is $k = \frac{y}{x}$ but closest is c. 5. Volume $V$ varies jointly as $r^2$ and $h$, so $V = kr^2h$. Answer: b. $V = kr^2h$. 6. Given $y$ varies directly as $x$, $y = kx$, and $y=15$ when $x=5$, so $k=\frac{15}{5}=3$. Find $y$ when $x=7$: $y=3 \times 7=21$. Answer: c. 21. 7. $y$ varies inversely as $x$: $y=\frac{k}{x}$. Given $y=5$, $k=60$, find $x$: $x=\frac{k}{y}=\frac{60}{5}=12$. Answer: a. 12. 8. Weight $W$ varies directly as mass $M$: $W=kM$. Given $W=24$ when $M=4$, $k=\frac{24}{4}=6$. 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 $m=2$, $k=10 \times 2=20$. 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 $h$: $A=kb h$. Given $A=12$ when $b=6$, $h=4$, find $k$: $k=\frac{12}{6 \times 4}=\frac{12}{24}=0.5$. New base $b=8$, new height $h=8$, new area: $A=0.5 \times 8 \times 8=32$. Answer: c. 32m^2. 11. Rule "add the exponents of the 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 $\frac{a^2 b^3 c^4 d^2 e^5}{a^4 c^7 d^8 e^3} = a^{2-4} b^3 c^{4-7} d^{2-8} e^{5-3} = a^{-2} b^3 c^{-3} d^{-6} e^{2}$. Rewrite with positive exponents: $\frac{b^3 e^2}{a^2 c^3 d^6}$. None of the options exactly match, but closest is c. 14. Evaluate $(x^5)^3 (x^3)^2 = x^{5 \times 3} x^{3 \times 2} = x^{15} x^{6} = x^{21}$. Options do not have $x^{21}$, so likely a typo; closest is a or b $x^{15}$. 15. Evaluate $(a^3)^{1/5} = a^{3 \times \frac{1}{5}} = a^{3/5}$. Answer: a. $a^{3/5}$. 16. Simplify $(a^1 b^{-1} c^0)^{5/2} = a^{5/2} b^{-5/2} c^0 = a^{5/2} b^{-5/2}$. Answer: a. $a^{15/2} b^{-5/2}$ is incorrect exponent on a, correct is $a^{5/2} b^{-5/2}$. Closest is a. 17. Equivalent radical of $2^{2/3} = \sqrt[3]{2^2} = \sqrt[3]{4}$. Answer: b. $\sqrt[3]{8}$ is $2^{3/3}=2$, so b is incorrect. Correct is $\sqrt[3]{4}$ which is not listed; closest is a or c. 18. Equivalent radical of $(2x)^{1/5} = \sqrt[5]{2x}$. Answer: none exactly match; a is $\sqrt[5]{32 x^5} = (2^5 x^5)^{1/5} = 2 x$. 19. Transform $\sqrt[n]{27 a^7} = (27 a^7)^{1/n} = 27^{1/n} a^{7/n}$. Answer: a. $27 a^{7/n}$ is incorrect, should be $27^{1/n} a^{7/n}$. Closest is a. 20. Equivalent to $\sqrt[8]{a^3} = a^{3/8}$. Answer: c. $2 a^{3/8}$ is incorrect, correct is $a^{3/8}$. 21. Simplify $7 \sqrt[18]{}$ is unclear; assuming $7 \sqrt[18]{...}$ missing. No clear answer. 22. Rationalize $\frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3}$. Answer: b. 23. Simplify $\sqrt[8]{a^3} = a^{3/8}$. Answer: none exactly match; closest is a. 24. Simplify $\sqrt[16]{a^5 b^3} = a^{5/16} b^{3/16}$. Answer: none exactly match. 25. Simplify $\sqrt[75/3]{}$ unclear; assuming $\sqrt{25} = 5$. Answer: a. 5. Final answers summarized for multiple choice: 1. b,d 2. a 3. a 4. none exact 5. b 6. c 7. a 8. a 9. c 10. c 11. c 12. a 13. c 14. none exact 15. a 16. a 17. none exact 18. none exact 19. a 20. none exact 22. b 25. a "slug": "variation exponents", "subject": "algebra", "desmos": {"latex": "", "features": {"intercepts": false, "extrema": false}}, "q_count": 25