1. Problem: If 18% of the goats crossed and that number is 45, find the total number of goats. Formula: Percentage crossed = $\frac{\text{number crossed}}{\text{total goats}} \times 100$. Calculation: $18 = \frac{45}{\text{total}} \times 100$. Solve for total: $\text{total} = \frac{45 \times 100}{18} = 250$ goats. 2. Problem: Find Jaime's weighted average with scores 75, 80, 88, 93 and weights 1, 2, 3, 4. Formula: Weighted average = $\frac{\sum (\text{score} \times \text{weight})}{\sum \text{weights}}$. Calculation: $\frac{75\times1 + 80\times2 + 88\times3 + 93\times4}{1+2+3+4} = \frac{75 + 160 + 264 + 372}{10} = \frac{871}{10} = 87.1$. 3. Problem: A 130% increase results in 1610 dolls. Find original number. Formula: New = Original + 130% of Original = Original $\times$ (1 + 1.3) = Original $\times$ 2.3. Calculation: $1610 = 2.3 \times \text{Original}$. Solve: $\text{Original} = \frac{1610}{2.3} = 700$ dolls. 4. Simplify $5\sqrt{80}$. $\sqrt{80} = \sqrt{16 \times 5} = 4\sqrt{5}$. So, $5\sqrt{80} = 5 \times 4\sqrt{5} = 20\sqrt{5}$. 5. Simplify $3\sqrt{120}$. $\sqrt{120} = \sqrt{4 \times 30} = 2\sqrt{30}$. So, $3\sqrt{120} = 3 \times 2\sqrt{30} = 6\sqrt{30}$. 6. Add $7\sqrt{5} - \sqrt{5} + 5\sqrt{3} - 3\sqrt{3}$. Group like terms: $(7 - 1)\sqrt{5} + (5 - 3)\sqrt{3} = 6\sqrt{5} + 2\sqrt{3}$. 8. Multiply $(3p - 4)(2p + 5)$. Use distributive property: $3p \times 2p = 6p^2$ $3p \times 5 = 15p$ $-4 \times 2p = -8p$ $-4 \times 5 = -20$ Sum: $6p^2 + (15p - 8p) - 20 = 6p^2 + 7p - 20$. 9. Graph $x = -\frac{1}{2}$ is a vertical line crossing x-axis at $-\frac{1}{2}$. 10. Graph $y = -\frac{1}{2}x$ is a line through origin with slope $-\frac{1}{2}$. 11. Graph $2y = x - 8$ or $y = \frac{x}{2} - 4$ is a line with slope $\frac{1}{2}$ and y-intercept $-4$. 12. Solve inequality $x + 3 > -7$ with domain positive integers. Subtract 3: $x > -10$. Since $x$ is positive integer, $x \geq 1$. 13. Solve system: $x + y = 10$ $-x + y = 0$ Add equations: $(x - x) + (y + y) = 10 + 0 \Rightarrow 2y = 10 \Rightarrow y = 5$. Substitute $y=5$ into $x + y = 10$: $x + 5 = 10 \Rightarrow x = 5$. 14. Solve system: $3x - 3y = 3$ $x - 5y = -3$ Multiply second by 3: $3x - 15y = -9$ Subtract first from this: $(3x - 15y) - (3x - 3y) = -9 - 3 \Rightarrow -12y = -12 \Rightarrow y = 1$. Substitute $y=1$ into $x - 5y = -3$: $x - 5 = -3 \Rightarrow x = 2$. 15. Solve system: $3x - y = 8$ $x - 3y = -8$ Multiply second by 3: $3x - 9y = -24$ Subtract first: $(3x - 9y) - (3x - y) = -24 - 8 \Rightarrow -8y = -32 \Rightarrow y = 4$. Substitute $y=4$ into $3x - y = 8$: $3x - 4 = 8 \Rightarrow 3x = 12 \Rightarrow x = 4$. 16. Simplify $\frac{a}{x} \div \frac{1}{a^2}$. Rewrite division as multiplication: $\frac{a}{x} \times \frac{a^2}{1} = \frac{a^3}{x}$. 17. Simplify $\frac{a}{a+b} \div a$. Rewrite: $\frac{a}{a+b} \times \frac{1}{a} = \frac{a}{a+b} \times \frac{1}{a} = \frac{\cancel{a}}{a+b} \times \frac{1}{\cancel{a}} = \frac{1}{a+b}$. 18. Simplify $\frac{x}{y} \div \frac{1}{y}$. Rewrite: $\frac{x}{y} \times \frac{y}{1} = x$. 19. Add $\frac{a}{x+y} + \frac{5}{x^2}$. No common denominator, so sum is $\frac{a}{x+y} + \frac{5}{x^2}$. 20. Add $1 + \frac{a}{b}$. Rewrite 1 as $\frac{b}{b}$: $\frac{b}{b} + \frac{a}{b} = \frac{a+b}{b}$. 21. Add $x + \frac{1}{x}$. No common denominator, sum is $x + \frac{1}{x}$. 22. Factor $4x^2 y^2 z - 8x^2 y^2 z^5$. Common factor: $4x^2 y^2 z$. Factor: $4x^2 y^2 z (1 - 2z^4)$. 23. Expand $-3x^{-2} y^2 \left( \frac{y^{-2}}{x^{-2}} + 4x^2 y \right)$. Rewrite inside parentheses: $\frac{y^{-2}}{x^{-2}} = y^{-2} x^{2}$. So expression: $-3x^{-2} y^2 (y^{-2} x^{2} + 4x^2 y)$. Distribute: $-3x^{-2} y^2 \times y^{-2} x^{2} = -3 x^{-2 + 2} y^{2 - 2} = -3 x^{0} y^{0} = -3$. $-3x^{-2} y^2 \times 4x^2 y = -12 x^{-2 + 2} y^{2 + 1} = -12 x^{0} y^{3} = -12 y^{3}$. Sum: $-3 - 12 y^{3}$. 24. Simplify $\frac{4kx - 4kx^2}{4kx}$. Factor numerator: $4kx(1 - x)$. Rewrite fraction: $\frac{4kx(1 - x)}{4kx}$. Cancel $4kx$: $\frac{\cancel{4kx}(1 - x)}{\cancel{4kx}} = 1 - x$. 25. Simplify $\frac{m^0 (p^{-2})^1 x^2 y^4}{(y^{-2})^2 y y^0 x^{-2}}$. Simplify powers: $m^0 = 1$, $(p^{-2})^1 = p^{-2}$, $(y^{-2})^2 = y^{-4}$, $y^0 = 1$. Rewrite numerator: $p^{-2} x^2 y^4$. Rewrite denominator: $y^{-4} y = y^{-4 + 1} = y^{-3} x^{-2}$. Combine denominator: $y^{-3} x^{-2}$. Rewrite fraction: $\frac{p^{-2} x^2 y^4}{y^{-3} x^{-2}} = p^{-2} x^{2 - (-2)} y^{4 - (-3)} = p^{-2} x^{4} y^{7}$. 26. Simplify $\left( \frac{(2x^{-2} y)^2}{p} \right) \left( \frac{p^2 x}{2} \right)^{-2}$. First part: $(2x^{-2} y)^2 = 4 x^{-4} y^{2}$. So first fraction: $\frac{4 x^{-4} y^{2}}{p}$. Second part: $\left( \frac{p^2 x}{2} \right)^{-2} = \left( \frac{2}{p^2 x} \right)^2 = \frac{4}{p^{4} x^{2}}$. Multiply: $\frac{4 x^{-4} y^{2}}{p} \times \frac{4}{p^{4} x^{2}} = \frac{16 x^{-4} y^{2}}{p^{5} x^{2}} = \frac{16 y^{2}}{p^{5} x^{6}}$. 27. Simplify $\frac{x^{2} x^{-3} e y^{2}}{y^{2} (x^{-4})^{2}}$. Simplify numerator: $x^{2 - 3} e y^{2} = x^{-1} e y^{2}$. Simplify denominator: $y^{2} x^{-8}$. Rewrite fraction: $\frac{x^{-1} e y^{2}}{y^{2} x^{-8}} = e x^{-1 - (-8)} y^{2 - 2} = e x^{7} y^{0} = e x^{7}$. 28. Simplify $\frac{3a^{2} x}{m} + \frac{5 x m^{-1}}{a^{-2}} - \frac{4 a a x^{-1}}{x^{-2} m}$. Rewrite terms: Second term denominator $a^{-2} = \frac{1}{a^{2}}$, so numerator $5 x m^{-1} = 5 x \frac{1}{m}$. So second term: $\frac{5 x / m}{1 / a^{2}} = 5 x / m \times a^{2} = \frac{5 a^{2} x}{m}$. Third term numerator: $4 a a x^{-1} = 4 a^{2} x^{-1}$. Denominator: $x^{-2} m = \frac{1}{x^{2}} m$. So third term: $\frac{4 a^{2} x^{-1}}{x^{-2} m} = 4 a^{2} x^{-1} \times \frac{x^{2}}{m} = \frac{4 a^{2} x^{1}}{m}$. Sum all: $\frac{3 a^{2} x}{m} + \frac{5 a^{2} x}{m} - \frac{4 a^{2} x}{m} = \frac{(3 + 5 - 4) a^{2} x}{m} = \frac{4 a^{2} x}{m}$. 29. Evaluate $-x^{-4} - x^{2} (x - m)$ for $x = -2$ and $m + 3 = 6$. Solve for $m$: $m = 3$. Calculate: $x^{-4} = (-2)^{-4} = \frac{1}{(-2)^4} = \frac{1}{16}$. $x^{2} = (-2)^2 = 4$. $x - m = -2 - 3 = -5$. So expression: $-\frac{1}{16} - 4 \times (-5) = -\frac{1}{16} + 20 = \frac{-1 + 320}{16} = \frac{319}{16}$. 30. Simplify $-3^{0} - 3(-2 - 2^{0})(-8^{0} - 5) + \sqrt{16}$. Calculate powers: $3^{0} = 1$, $2^{0} = 1$, $-8^{0} = -1$ (since $8^{0} = 1$, so $-8^{0} = -1$). Simplify inside parentheses: $-2 - 1 = -3$. $-1 - 5 = -6$. Calculate: $-1 - 3 \times (-3) \times (-6) + 4$. $-1 - 3 \times 18 + 4 = -1 - 54 + 4 = -51$. Final answers: 1. 250 goats 2. 87.1 3. 700 dolls 4. $20\sqrt{5}$ 5. $6\sqrt{30}$ 6. $6\sqrt{5} + 2\sqrt{3}$ 8. $6p^{2} + 7p - 20$ 9. Vertical line $x = -\frac{1}{2}$ 10. Line $y = -\frac{1}{2}x$ 11. Line $y = \frac{x}{2} - 4$ 12. $x \geq 1$ (positive integers) 13. $(x,y) = (5,5)$ 14. $(x,y) = (2,1)$ 15. $(x,y) = (4,4)$ 16. $\frac{a^{3}}{x}$ 17. $\frac{1}{a+b}$ 18. $x$ 19. $\frac{a}{x+y} + \frac{5}{x^{2}}$ 20. $\frac{a+b}{b}$ 21. $x + \frac{1}{x}$ 22. $4x^{2} y^{2} z (1 - 2 z^{4})$ 23. $-3 - 12 y^{3}$ 24. $1 - x$ 25. $p^{-2} x^{4} y^{7}$ 26. $\frac{16 y^{2}}{p^{5} x^{6}}$ 27. $e x^{7}$ 28. $\frac{4 a^{2} x}{m}$ 29. $\frac{319}{16}$ 30. $-51$