Multiply Rational Expressions

1. **Problem:** Multiply $$\frac{9p}{7s} \times \frac{28s}{81p}$$. 2. **Formula:** Multiply numerators and denominators: $$\frac{a}{b} \times \frac{c}{d} = \frac{ac}{bd}$$. 3. **Work:** $$\frac{9p \times 28s}{7s \times 81p} = \frac{252ps}{567ps}$$. 4. **Simplify:** Cancel common factors $p$ and $s$: $$\frac{252}{567} = \frac{28}{63} = \frac{4}{9}$$. 5. **Answer:** $$\frac{4}{9}$$. --- 1. **Problem:** Multiply $$\frac{2a^2 b^2}{9c^2 d^2} \times \frac{45cd}{14ab}$$. 2. **Formula:** Multiply numerators and denominators. 3. **Work:** $$\frac{2a^2 b^2 \times 45cd}{9c^2 d^2 \times 14ab} = \frac{90 a^2 b^2 c d}{126 a b c^2 d^2}$$. 4. **Simplify:** Cancel $a$, $b$, $c$, $d$: $$\frac{90 a^{2-1} b^{2-1} c^{1-2} d^{1-2}}{126} = \frac{90 a b c^{-1} d^{-1}}{126} = \frac{90 a b}{126 c d}$$. 5. **Reduce fraction:** $$\frac{90}{126} = \frac{15}{21} = \frac{5}{7}$$. 6. **Answer:** $$\frac{5 a b}{7 c d}$$. --- 1. **Problem:** Multiply $$\frac{3m - 33}{m + 10} \times \frac{2m + 20}{8m - 88}$$. 2. **Factor:** $$3m - 33 = 3(m - 11)$$ $$2m + 20 = 2(m + 10)$$ $$8m - 88 = 8(m - 11)$$ 3. **Rewrite:** $$\frac{3(m - 11)}{m + 10} \times \frac{2(m + 10)}{8(m - 11)}$$. 4. **Multiply:** $$\frac{3(m - 11) \times 2(m + 10)}{(m + 10) \times 8(m - 11)} = \frac{6 (m - 11)(m + 10)}{8 (m + 10)(m - 11)}$$. 5. **Cancel:** $(m - 11)$ and $(m + 10)$ cancel. 6. **Simplify:** $$\frac{6}{8} = \frac{3}{4}$$. 7. **Answer:** $$\frac{3}{4}$$. --- 1. **Problem:** Multiply $$\frac{-20w}{15 - 5w} \times \frac{25w - 75}{w + 2}$$. 2. **Factor:** $$15 - 5w = 5(3 - w)$$ $$25w - 75 = 25(w - 3)$$ 3. **Rewrite:** $$\frac{-20w}{5(3 - w)} \times \frac{25(w - 3)}{w + 2}$$. 4. **Note:** $3 - w = -(w - 3)$, so denominator is $5(3 - w) = -5(w - 3)$. 5. **Rewrite denominator:** $$\frac{-20w}{-5(w - 3)} \times \frac{25(w - 3)}{w + 2} = \frac{-20w}{-5(w - 3)} \times \frac{25(w - 3)}{w + 2}$$. 6. **Simplify:** $$\frac{-20w}{-5(w - 3)} = \frac{20w}{5(w - 3)} = \frac{4w}{w - 3}$$. 7. **Multiply:** $$\frac{4w}{w - 3} \times \frac{25(w - 3)}{w + 2} = \frac{4w \times 25 (w - 3)}{(w - 3)(w + 2)}$$. 8. **Cancel:** $(w - 3)$ cancels. 9. **Result:** $$\frac{100 w}{w + 2}$$. 10. **Answer:** $$\frac{100 w}{w + 2}$$. --- 1. **Problem:** Multiply $$\frac{y^2 - 36}{y + 6} \times \frac{y}{y - 6}$$. 2. **Factor:** $$y^2 - 36 = (y - 6)(y + 6)$$. 3. **Rewrite:** $$\frac{(y - 6)(y + 6)}{y + 6} \times \frac{y}{y - 6}$$. 4. **Cancel:** $(y + 6)$ cancels. 5. **Multiply:** $$\frac{y - 6}{1} \times \frac{y}{y - 6}$$. 6. **Cancel:** $(y - 6)$ cancels. 7. **Answer:** $$y$$. --- 1. **Problem:** Multiply $$\frac{6a + 12}{a^2 - a} \times \frac{4a^3 - 4a^2}{a^2 - 4}$$. 2. **Factor:** $$6a + 12 = 6(a + 2)$$ $$a^2 - a = a(a - 1)$$ $$4a^3 - 4a^2 = 4a^2(a - 1)$$ $$a^2 - 4 = (a - 2)(a + 2)$$ 3. **Rewrite:** $$\frac{6(a + 2)}{a(a - 1)} \times \frac{4a^2 (a - 1)}{(a - 2)(a + 2)}$$. 4. **Multiply:** $$\frac{6(a + 2) \times 4a^2 (a - 1)}{a(a - 1)(a - 2)(a + 2)}$$. 5. **Cancel:** $(a + 2)$ and $(a - 1)$ cancel. 6. **Simplify:** $$\frac{6 \times 4 a^2}{a (a - 2)} = \frac{24 a^2}{a (a - 2)} = \frac{24 a}{a - 2}$$. 7. **Answer:** $$\frac{24 a}{a - 2}$$. --- 1. **Problem:** Multiply $$\frac{9p^2 - 12p + 4}{p^2 - 4p - 32} \times \frac{p^2 + 4p}{3p^4 - 2p^3}$$. 2. **Factor:** $$9p^2 - 12p + 4 = (3p - 2)^2$$ $$p^2 - 4p - 32 = (p - 8)(p + 4)$$ $$p^2 + 4p = p(p + 4)$$ $$3p^4 - 2p^3 = p^3(3p - 2)$$ 3. **Rewrite:** $$\frac{(3p - 2)^2}{(p - 8)(p + 4)} \times \frac{p(p + 4)}{p^3 (3p - 2)}$$. 4. **Multiply:** $$\frac{(3p - 2)^2 p (p + 4)}{(p - 8)(p + 4) p^3 (3p - 2)}$$. 5. **Cancel:** $(p + 4)$ and one $(3p - 2)$ and $p$ from numerator and denominator. 6. **Simplify:** $$\frac{(3p - 2) p^{1} }{(p - 8) p^{3}} = \frac{3p - 2}{(p - 8) p^{2}}$$. 7. **Answer:** $$\frac{3p - 2}{p^{2} (p - 8)}$$. --- 1. **Problem:** Multiply $$\frac{b^3 + 125}{2b^2 + 9b - 5} \times \frac{4b^4 - 4b^2}{2b^3 - 10b^2 + 50b}$$. 2. **Factor:** $$b^3 + 125 = (b + 5)(b^2 - 5b + 25)$$ $$2b^2 + 9b - 5 = (2b - 1)(b + 5)$$ $$4b^4 - 4b^2 = 4b^2(b^2 - 1) = 4b^2 (b - 1)(b + 1)$$ $$2b^3 - 10b^2 + 50b = 2b(b^2 - 5b + 25)$$ 3. **Rewrite:** $$\frac{(b + 5)(b^2 - 5b + 25)}{(2b - 1)(b + 5)} \times \frac{4b^2 (b - 1)(b + 1)}{2b (b^2 - 5b + 25)}$$. 4. **Multiply:** $$\frac{(b + 5)(b^2 - 5b + 25) 4b^2 (b - 1)(b + 1)}{(2b - 1)(b + 5) 2b (b^2 - 5b + 25)}$$. 5. **Cancel:** $(b + 5)$ and $(b^2 - 5b + 25)$. 6. **Simplify:** $$\frac{4b^2 (b - 1)(b + 1)}{(2b - 1) 2b} = \frac{4b^2 (b - 1)(b + 1)}{2b (2b - 1)}$$. 7. **Reduce:** $$\frac{4b^2}{2b} = 2b$$. 8. **Answer:** $$\frac{2b (b - 1)(b + 1)}{2b - 1}$$. --- 1. **Problem:** Multiply $$\frac{4d^2 - 1}{d^2 + 4d - 5} \times \frac{2d^2 - 2d}{4d^3 - 2d^2 - 2d}$$. 2. **Factor:** $$4d^2 - 1 = (2d - 1)(2d + 1)$$ $$d^2 + 4d - 5 = (d + 5)(d - 1)$$ $$2d^2 - 2d = 2d(d - 1)$$ $$4d^3 - 2d^2 - 2d = 2d(2d^2 - d - 1) = 2d(2d + 1)(d - 1)$$ 3. **Rewrite:** $$\frac{(2d - 1)(2d + 1)}{(d + 5)(d - 1)} \times \frac{2d (d - 1)}{2d (2d + 1)(d - 1)}$$. 4. **Cancel:** $2d$, $(d - 1)$, and $(2d + 1)$. 5. **Result:** $$\frac{2d - 1}{d + 5}$$. 6. **Answer:** $$\frac{2d - 1}{d + 5}$$. --- 1. **Problem:** Multiply $$\frac{2n^2 + 3n - 2}{n^3 + 4n^2 + n - 6} \times \frac{3n^3 + 6n^2 - 9n}{n^3 + 8}$$. 2. **Factor:** $$2n^2 + 3n - 2 = (2n - 1)(n + 2)$$ $$n^3 + 4n^2 + n - 6 = (n + 3)(n^2 + n - 2) = (n + 3)(n + 2)(n - 1)$$ $$3n^3 + 6n^2 - 9n = 3n(n^2 + 2n - 3) = 3n(n + 3)(n - 1)$$ $$n^3 + 8 = (n + 2)(n^2 - 2n + 4)$$ 3. **Rewrite:** $$\frac{(2n - 1)(n + 2)}{(n + 3)(n + 2)(n - 1)} \times \frac{3n (n + 3)(n - 1)}{(n + 2)(n^2 - 2n + 4)}$$. 4. **Cancel:** $(n + 2)$, $(n + 3)$, and $(n - 1)$. 5. **Result:** $$\frac{(2n - 1) \times 3n}{n^2 - 2n + 4} = \frac{3n (2n - 1)}{n^2 - 2n + 4}$$. 6. **Answer:** $$\frac{3n (2n - 1)}{n^2 - 2n + 4}$$. --- **Summary:** Each problem involves multiplying rational expressions by factoring numerators and denominators, canceling common factors, and simplifying.