1. **Multiply and simplify** A1. Multiply \(\frac{9r^3 - 54r^2}{9r^2 + 45r} \cdot \frac{9r^2 + 9r}{9r^3 - 54r^2}\). Factor each expression: \(9r^3 - 54r^2 = 9r^2(r - 6)\) \(9r^2 + 45r = 9r(r + 5)\) \(9r^2 + 9r = 9r(r + 1)\) \(9r^3 - 54r^2 = 9r^2(r - 6)\) The product is: $$ \frac{9r^2(r - 6)}{9r(r + 5)} \cdot \frac{9r(r + 1)}{9r^2(r - 6)} $$ Cancel common factors: \(9r^2\), \(9r\), and \((r - 6)\) cancel out. Leftover: $$ \frac{1}{r + 5} \cdot \frac{r + 1}{1} = \frac{r + 1}{r + 5} $$ A2. Multiply \(\frac{5r + 50}{r + 10} \cdot \frac{r - 2}{5}\). Factor numerator: \(5r + 50 = 5(r + 10)\), so: $$ \frac{5(r + 10)}{r + 10} \cdot \frac{r - 2}{5} $$ Cancel \(5\) and \(r + 10\): $$ 1 \cdot (r - 2) = r - 2 $$ 2. **Divide and simplify** B1. Divide \(\frac{3}{28b} \div \frac{3}{b + 1}\). Dividing by a fraction is multiplying by its reciprocal: $$ \frac{3}{28b} \cdot \frac{b + 1}{3} $$ Cancel \(3\): $$ \frac{b + 1}{28b} $$ B2. Divide \(\frac{4n}{n - 6} \div \frac{4n}{8n - 48}\). Write reciprocal of second: $$ \frac{4n}{n - 6} \cdot \frac{8n - 48}{4n} $$ Factor \(8n - 48 = 8(n - 6)\): $$ \frac{4n}{n - 6} \cdot \frac{8(n - 6)}{4n} $$ Cancel \(4n\) and \(n - 6\): $$ 8 $$ 3. **Perform indicated operations and simplify** C1. \(\frac{7x}{x - 6} + \frac{42}{6 - x}\) Note \(6 - x = -(x - 6)\), so rewrite second fraction: $$ \frac{7x}{x - 6} - \frac{42}{x - 6} = \frac{7x - 42}{x - 6} = \frac{7(x - 6)}{x - 6} = 7 $$ C2. \(\frac{14}{15 - 5x} - \frac{8}{2x - 6}\) Factor denominators: \(15 - 5x = 5(3 - x)\), \(2x - 6 = 2(x - 3)\). Rewrite \(3 - x = -(x - 3)\), so \(15 - 5x = -5(x - 3)\). Both denominators have \(x - 3\) up to a sign. Rewrite expression: $$ \frac{14}{-5(x - 3)} - \frac{8}{2(x - 3)} = -\frac{14}{5(x - 3)} - \frac{8}{2(x - 3)} $$ Common denominator: \(10(x - 3)\). Rewrite fractions: $$ -\frac{14 \cdot 2}{10(x - 3)} - \frac{8 \cdot 5}{10(x - 3)} = \frac{-28 - 40}{10(x - 3)} = \frac{-68}{10(x - 3)} = \frac{-34}{5(x - 3)} $$ C3. \(\frac{x}{x + 2} - \frac{3x}{4x - 1}\) Common denominator: \((x + 2)(4x - 1)\). Rewrite: $$ \frac{x(4x - 1)}{(x + 2)(4x - 1)} - \frac{3x(x + 2)}{(x + 2)(4x - 1)} = \frac{4x^2 - x - 3x^2 - 6x}{(x + 2)(4x - 1)} = \frac{x^2 - 7x}{(x + 2)(4x - 1)} $$ Factor numerator: $$ x(x - 7) $$ So: $$ \frac{x(x - 7)}{(x + 2)(4x - 1)} $$ C4. \(\frac{4z}{z^2 + z - 20} + \frac{z}{z^2 - 8z + 16}\) Factor denominators: \(z^2 + z - 20 = (z + 5)(z - 4)\) \(z^2 - 8z + 16 = (z - 4)^2\) Common denominator: \((z + 5)(z - 4)^2\) Rewrite fractions: $$ \frac{4z (z - 4)}{(z + 5)(z - 4)^2} + \frac{z (z + 5)}{(z + 5)(z - 4)^2} = \frac{4z (z - 4) + z (z + 5)}{(z + 5)(z - 4)^2} $$ Simplify numerator: $$ 4z^2 -16z + z^2 + 5z = 5z^2 -11z $$ Final: $$ \frac{5z^2 - 11z}{(z + 5)(z - 4)^2} = \frac{z(5z - 11)}{(z + 5)(z - 4)^2} $$ 4. **Add and simplify each expression** (1) \(\frac{x + 5}{x + 4} + \frac{3 + x}{x + 4}\) add numerators (notice \(3 + x = x + 3\)): $$ \frac{x + 5 + x + 3}{x + 4} = \frac{2x + 8}{x + 4} = \frac{2(x + 4)}{x + 4} = 2 $$ (2) \(\frac{5x + 4}{2x + 3} + \frac{x + 2}{2x + 3}\): $$ \frac{5x + 4 + x + 2}{2x + 3} = \frac{6x + 6}{2x + 3} = \frac{6(x + 1)}{2x + 3} $$ (cannot simplify further) (3) \(\frac{2x + 4}{8 + 3x} + \frac{x + 7}{8 + 3x}\) add numerators: $$ \frac{2x + 4 + x + 7}{8 + 3x} = \frac{3x + 11}{8 + 3x} $$ No simple factorization or simplification. (4) \(\frac{2 + x}{6x + 5} + \frac{4x + 1}{6x + 5}\) add numerators: $$ \frac{2 + x + 4x + 1}{6x + 5} = \frac{5x + 3}{6x + 5} $$ 5. **Determine TRUE or FALSE statements** 1. \(\frac{8x^2 y^2}{9z^5}\) simplest form? TRUE (no common factors) 2. \(\frac{x^2 - 9}{x - 3}\) simplest form? FALSE (can factor numerator as \((x - 3)(x + 3)\)) 3. Complete factorization of \(\frac{x^2 - 16}{x - 4}\) is \(\frac{(x - 4)(x + 4)}{x - 4}\)? TRUE 4. Complete factorization of \(\frac{x + 3}{x^3 + 27}\) is \(\frac{(x + 3)}{(x + 3)(x^2 + 3x + 9)}\)? TRUE 5. \(\frac{x^3 - 8}{x + 2}\) simplest form? TRUE (difference of cubes factored as \((x - 2)(x^2 + 2x + 4)\); but denominator is \(x + 2\), which does not factor numerator, so simplest) 6. \(\frac{6x + 18}{9x - 3}\) simplest form? FALSE (can factor numerator as \(6(x + 3)\) and denominator as \(3(3x - 1)\); no common factors but numbers can factor) 7. Complete factorization of \(\frac{x^2 - 8x + 15}{x - 3}\) is \(\frac{(x - 3)(x - 5)}{x - 3}\)? TRUE 8. Complete factorization of \(\frac{x^2 + x - 6}{x + 3}\) is \(\frac{(x + 3)(x - 2)}{x + 3}\)? TRUE 9. \(\frac{m^3 - 16}{m^2 - 4}\) simplest form? FALSE (numerator is difference of cubes except 16 is not a cube; denominator is difference of squares) 10. \(\frac{x^2 - 5x + 6}{x^2 - 3x - 10}\) simplest form? FALSE (both numerator and denominator can be factored) Final answers summarized in each step above.