Quotient Simplify 5567Ec

1. **Problem statement:** Find the quotient and simplify for problems 28 to 41. 2. **Recall division of fractions:** Dividing by a fraction is the same as multiplying by its reciprocal. --- **28.** $\frac{4}{5} \div \frac{3}{10}$ $$= \frac{4}{5} \times \frac{10}{3}$$ $$= \frac{4 \times 10}{5 \times 3} = \frac{40}{15}$$ Simplify by dividing numerator and denominator by 5: $$= \frac{\cancel{40}^{8}}{\cancel{15}^{3}} = \frac{8}{3}$$ --- **29.** $\frac{6c^{2}}{(4c)^{2}} \div \frac{90c^{5}}{16c^{6}}$ First, expand denominator: $$(4c)^{2} = 16c^{2}$$ So expression is: $$\frac{6c^{2}}{16c^{2}} \div \frac{90c^{5}}{16c^{6}}$$ Rewrite division as multiplication by reciprocal: $$= \frac{6c^{2}}{16c^{2}} \times \frac{16c^{6}}{90c^{5}}$$ Simplify factors: Cancel $c^{2}$ in numerator and denominator of first fraction: $$= \frac{6}{16} \times \frac{16c^{6}}{90c^{5}}$$ Cancel 16 in numerator and denominator: $$= \frac{6}{\cancel{16}} \times \frac{\cancel{16}c^{6}}{90c^{5}} = \frac{6}{1} \times \frac{c^{6}}{90c^{5}}$$ Simplify $c^{6}/c^{5} = c^{1} = c$: $$= \frac{6}{1} \times \frac{c}{90} = \frac{6c}{90}$$ Simplify $6/90$ by dividing numerator and denominator by 6: $$= \frac{c}{15}$$ --- **30.** $\frac{15e}{f} \div \frac{30e}{f^{2}}$ Rewrite division as multiplication by reciprocal: $$= \frac{15e}{f} \times \frac{f^{2}}{30e}$$ Cancel $e$ in numerator and denominator: $$= \frac{15}{f} \times \frac{f^{2}}{30}$$ Multiply numerators and denominators: $$= \frac{15 f^{2}}{30 f}$$ Simplify $15/30 = 1/2$ and $f^{2}/f = f$: $$= \frac{1 \times f}{2} = \frac{f}{2}$$ --- **31.** $\frac{3j^{2}}{5k} \div \frac{6j}{10k}$ Rewrite division as multiplication by reciprocal: $$= \frac{3j^{2}}{5k} \times \frac{10k}{6j}$$ Cancel $k$ in numerator and denominator: $$= \frac{3j^{2}}{5} \times \frac{10}{6j}$$ Multiply numerators and denominators: $$= \frac{3j^{2} \times 10}{5 \times 6j} = \frac{30j^{2}}{30j}$$ Simplify numerator and denominator by 30: $$= \frac{\cancel{30}j^{2}}{\cancel{30}j} = \frac{j^{2}}{j} = j$$ --- **32.** $\frac{6pq}{4} \div \frac{q^{2}}{12}$ Rewrite division as multiplication by reciprocal: $$= \frac{6pq}{4} \times \frac{12}{q^{2}}$$ Multiply numerators and denominators: $$= \frac{6pq \times 12}{4 \times q^{2}} = \frac{72 p q}{4 q^{2}}$$ Simplify $72/4 = 18$ and $q/q^{2} = 1/q$: $$= 18 p \times \frac{1}{q} = \frac{18 p}{q}$$ --- **33.** $\frac{t^{4} - u^{4}}{4t + 4u} \div \frac{4t^{2} + 4u^{2}}{16}$ Rewrite division as multiplication by reciprocal: $$= \frac{t^{4} - u^{4}}{4t + 4u} \times \frac{16}{4t^{2} + 4u^{2}}$$ Factor numerator and denominator: $t^{4} - u^{4} = (t^{2} - u^{2})(t^{2} + u^{2}) = (t - u)(t + u)(t^{2} + u^{2})$ $4t + 4u = 4(t + u)$ $4t^{2} + 4u^{2} = 4(t^{2} + u^{2})$ Substitute: $$= \frac{(t - u)(t + u)(t^{2} + u^{2})}{4(t + u)} \times \frac{16}{4(t^{2} + u^{2})}$$ Cancel $t + u$ and $t^{2} + u^{2}$: $$= \frac{(t - u)\cancel{(t + u)}\cancel{(t^{2} + u^{2})}}{4\cancel{(t + u)}} \times \frac{16}{4\cancel{(t^{2} + u^{2})}} = \frac{t - u}{4} \times \frac{16}{4}$$ Multiply: $$= \frac{(t - u) \times 16}{4 \times 4} = \frac{16(t - u)}{16}$$ Simplify: $$= t - u$$ --- **34.** $\frac{a^{2} + a - 12}{a^{2} + 3a - 4} \div \frac{3 - a}{1 - a}$ Rewrite division as multiplication by reciprocal: $$= \frac{a^{2} + a - 12}{a^{2} + 3a - 4} \times \frac{1 - a}{3 - a}$$ Factor quadratics: $a^{2} + a - 12 = (a + 4)(a - 3)$ $a^{2} + 3a - 4 = (a + 4)(a - 1)$ Rewrite $1 - a = -(a - 1)$ and $3 - a = -(a - 3)$ Substitute: $$= \frac{(a + 4)(a - 3)}{(a + 4)(a - 1)} \times \frac{-(a - 1)}{-(a - 3)}$$ Negatives cancel: $$= \frac{(a + 4)(a - 3)}{(a + 4)(a - 1)} \times \frac{(a - 1)}{(a - 3)}$$ Cancel $(a + 4)$, $(a - 3)$, and $(a - 1)$: $$= 1$$ --- **35.** $\frac{2c^{2} + 7c - 15}{27 - 18c} \div \frac{c^{4} + c^{3} - 20c^{2}}{6c^{3} - 96c}$ Rewrite division as multiplication by reciprocal: $$= \frac{2c^{2} + 7c - 15}{27 - 18c} \times \frac{6c^{3} - 96c}{c^{4} + c^{3} - 20c^{2}}$$ Factor each polynomial: $2c^{2} + 7c - 15 = (2c - 3)(c + 5)$ $27 - 18c = 9(3 - 2c)$ $6c^{3} - 96c = 6c(c^{2} - 16) = 6c(c - 4)(c + 4)$ $c^{4} + c^{3} - 20c^{2} = c^{2}(c^{2} + c - 20) = c^{2}(c + 5)(c - 4)$ Rewrite $3 - 2c = -(2c - 3)$ Substitute: $$= \frac{(2c - 3)(c + 5)}{9 \times -(2c - 3)} \times \frac{6c (c - 4)(c + 4)}{c^{2} (c + 5)(c - 4)}$$ Simplify negatives: $$= \frac{(2c - 3)(c + 5)}{-9 (2c - 3)} \times \frac{6c (c - 4)(c + 4)}{c^{2} (c + 5)(c - 4)}$$ Cancel $(2c - 3)$, $(c + 5)$, and $(c - 4)$: $$= \frac{1}{-9} \times \frac{6c (c + 4)}{c^{2}} = -\frac{1}{9} \times \frac{6c (c + 4)}{c^{2}}$$ Simplify $c/c^{2} = 1/c$: $$= -\frac{1}{9} \times \frac{6 (c + 4)}{c} = -\frac{6 (c + 4)}{9 c}$$ Simplify $6/9 = 2/3$: $$= -\frac{2 (c + 4)}{3 c}$$ --- **36.** $\frac{1}{e^{3}f^{3}} \div \frac{1}{ef^{2}} \div \frac{1}{e^{2}f}$ Rewrite as: $$\left(\frac{1}{e^{3}f^{3}} \div \frac{1}{ef^{2}}\right) \div \frac{1}{e^{2}f}$$ First division: $$= \frac{1}{e^{3}f^{3}} \times \frac{ef^{2}}{1} = \frac{ef^{2}}{e^{3}f^{3}}$$ Simplify $e/e^{3} = 1/e^{2}$ and $f^{2}/f^{3} = 1/f$: $$= \frac{1}{e^{2} f}$$ Now divide by $\frac{1}{e^{2} f}$: $$= \frac{1}{e^{2} f} \div \frac{1}{e^{2} f} = 1$$ --- **37.** $\frac{4 + 4j}{30} \div \frac{1 + j}{45}$ Rewrite division as multiplication by reciprocal: $$= \frac{4 + 4j}{30} \times \frac{45}{1 + j}$$ Factor numerator: $$4 + 4j = 4(1 + j)$$ Substitute: $$= \frac{4(1 + j)}{30} \times \frac{45}{1 + j}$$ Cancel $(1 + j)$: $$= \frac{4}{30} \times 45 = \frac{4 \times 45}{30}$$ Simplify $45/30 = 3/2$: $$= 4 \times \frac{3}{2} = 6$$ --- **38.** $\frac{n^{2} - 16}{3} \div \frac{n + 4}{21}$ Rewrite division as multiplication by reciprocal: $$= \frac{n^{2} - 16}{3} \times \frac{21}{n + 4}$$ Factor numerator: $$n^{2} - 16 = (n - 4)(n + 4)$$ Substitute: $$= \frac{(n - 4)(n + 4)}{3} \times \frac{21}{n + 4}$$ Cancel $(n + 4)$: $$= \frac{n - 4}{3} \times 21 = (n - 4) \times \frac{21}{3} = 7(n - 4)$$ --- **39.** $\frac{8}{r - s} \div \frac{8}{s - r}$ Rewrite division as multiplication by reciprocal: $$= \frac{8}{r - s} \times \frac{s - r}{8}$$ Simplify $8/8 = 1$: $$= \frac{s - r}{r - s}$$ Note that $s - r = -(r - s)$, so: $$= \frac{-(r - s)}{r - s} = -1$$ --- **40.** $\frac{u^{3} - v^{3}}{4u^{2}v} \div (u - v)$ Rewrite division as multiplication by reciprocal: $$= \frac{u^{3} - v^{3}}{4u^{2}v} \times \frac{1}{u - v} = \frac{u^{3} - v^{3}}{4u^{2}v (u - v)}$$ Factor numerator: $$u^{3} - v^{3} = (u - v)(u^{2} + uv + v^{2})$$ Substitute: $$= \frac{(u - v)(u^{2} + uv + v^{2})}{4u^{2}v (u - v)}$$ Cancel $(u - v)$: $$= \frac{u^{2} + uv + v^{2}}{4u^{2}v}$$ --- **41.** $\left(\frac{2x - 5}{12}\right)^{2} \div \left(\frac{5 - 2x}{4}\right)^{4}$ Rewrite division as multiplication by reciprocal: $$= \left(\frac{2x - 5}{12}\right)^{2} \times \left(\frac{4}{5 - 2x}\right)^{4}$$ Note that $5 - 2x = -(2x - 5)$, so: $$\left(\frac{4}{5 - 2x}\right)^{4} = \left(\frac{4}{-(2x - 5)}\right)^{4} = \left(-\frac{4}{2x - 5}\right)^{4} = \left(\frac{4}{2x - 5}\right)^{4}$$ Since even power, negative sign disappears. Rewrite expression: $$= \left(\frac{2x - 5}{12}\right)^{2} \times \left(\frac{4}{2x - 5}\right)^{4} = \frac{(2x - 5)^{2}}{12^{2}} \times \frac{4^{4}}{(2x - 5)^{4}}$$ Combine powers of $(2x - 5)$: $$= \frac{4^{4}}{12^{2}} \times \frac{(2x - 5)^{2}}{(2x - 5)^{4}} = \frac{4^{4}}{12^{2}} \times (2x - 5)^{2 - 4} = \frac{4^{4}}{12^{2}} \times (2x - 5)^{-2}$$ Calculate powers: $$4^{4} = 256, \quad 12^{2} = 144$$ Rewrite: $$= \frac{256}{144} \times \frac{1}{(2x - 5)^{2}} = \frac{256}{144 (2x - 5)^{2}}$$ Simplify $256/144$ by dividing numerator and denominator by 16: $$= \frac{16}{9 (2x - 5)^{2}}$$ --- **Final answers:** 28. $\frac{8}{3}$ 29. $\frac{c}{15}$ 30. $\frac{f}{2}$ 31. $j$ 32. $\frac{18p}{q}$ 33. $t - u$ 34. $1$ 35. $-\frac{2(c + 4)}{3c}$ 36. $1$ 37. $6$ 38. $7(n - 4)$ 39. $-1$ 40. $\frac{u^{2} + uv + v^{2}}{4u^{2}v}$ 41. $\frac{16}{9(2x - 5)^{2}}$