1. Выполните умножение или деление: а) \(\frac{a}{3b} \cdot \frac{5b}{7a}\) б) \(\frac{x^3}{6y^{10}} : \frac{3y^9}{x^7}\) в) \(\frac{3a^2b}{c}\) : \(\frac{c}{a^2b}\) г) \(\frac{2m-n}{3p} \cdot \frac{3}{2m-n}\) д) \(\frac{m^2-mn}{p^2+pq} \cdot \frac{p+q}{m-n}\) е) \(\frac{3a^{11}}{5b^{15}} : \frac{21a^{10}}{10b^{14}}\) ж) \(\frac{m-n}{p+q} \cdot \frac{2p+2q}{3m-3n}\) з) \(\frac{mn-n^2}{pq+q^2} : \frac{n^2-mn}{q^2-nq}\) и) \(\frac{8a^2b}{c} \cdot \frac{a^2b}{8c}\) к) \(\frac{5a}{3c} : \frac{3a}{5c}\) л) \(\frac{5b}{a^2-b^2} \cdot (a+b)\) м) \(\frac{a^2-b^2}{x+3y} : \frac{a-b}{x+3y}\) н) \(\frac{a^2-b^2}{x+y} \cdot \frac{x-y}{a+b}\) о) \(\frac{a^2-b^2}{x+y} : \frac{a+b}{x+y}\) п) \(\frac{x^2+2xy+y^2}{a^2-b^2}\) 3) а) \(\frac{a^2-9b^2}{c^2+8cd+16d^2} \cdot \frac{c^2-16d^2}{3b-a}\) б) \(\frac{a^2-b^2+a+b}{x^2-y^2+x-y} \cdot \frac{3a+3b}{2x-2y}\)

Решение:

1. а) \(\frac{a}{3b} \cdot \frac{5b}{7a} = \frac{a · 5b}{3b · 7a} = \frac{5ab}{21ab} = \frac{5}{21}\)

1. б) \(\frac{x^3}{6y^{10}} : \frac{3y^9}{x^7} = \frac{x^3}{6y^{10}} \cdot \frac{x^7}{3y^9} = \frac{x^{10}}{18y^{19}}\)

1. в) \(\frac{3a^2b}{c} : \frac{c}{a^2b} = \frac{3a^2b}{c} \cdot \frac{a^2b}{c} = \frac{3a^4b^2}{c^2}\)

1. г) \(\frac{2m-n}{3p} \cdot \frac{3}{2m-n} = \frac{(2m-n) · 3}{3p · (2m-n)} = \frac{1}{p}\)

1. д) \(\frac{m^2-mn}{p^2+pq} \cdot \frac{p+q}{m-n} = \frac{m(m-n)}{p(p+q)} \cdot \frac{p+q}{m-n} = \frac{m}{p}\)

1. е) \(\frac{3a^{11}}{5b^{15}} : \frac{21a^{10}}{10b^{14}} = \frac{3a^{11}}{5b^{15}} \cdot \frac{10b^{14}}{21a^{10}} = \frac{3 · 10 · a^{11} · b^{14}}{5 · 21 · b^{15} · a^{10}} = \frac{30a}{105b} = \frac{2a}{7b}\)

1. ж) \(\frac{m-n}{p+q} \cdot \frac{2p+2q}{3m-3n} = \frac{m-n}{p+q} \cdot \frac{2(p+q)}{3(m-n)} = \frac{2}{3}\)

1. з) \(\frac{mn-n^2}{pq+q^2} : \frac{n^2-mn}{q^2-nq} = \frac{n(m-n)}{q(p+q)} : \frac{n(n-m)}{q(q-n)} = \frac{n(m-n)}{q(p+q)} \cdot \frac{q(q-n)}{n(n-m)} = \frac{m-n}{p+q} \cdot \frac{q-n}{n-m} = \frac{m-n}{p+q} \cdot \frac{-(n-q)}{n-m} = -\frac{m-n}{p+q}\)

1. и) \(\frac{8a^2b}{c} \cdot \frac{a^2b}{8c} = \frac{8a^4b^2}{8c^2} = \frac{a^4b^2}{c^2}\)

1. к) \(\frac{5a}{3c} : \frac{3a}{5c} = \frac{5a}{3c} \cdot \frac{5c}{3a} = \frac{25ac}{9ac} = \frac{25}{9}\)

1. л) \(\frac{5b}{a^2-b^2} \cdot (a+b) = \frac{5b}{(a-b)(a+b)} · (a+b) = \frac{5b}{a-b}\)

1. м) \(\frac{a^2-b^2}{x+3y} : \frac{a-b}{x+3y} = \frac{a^2-b^2}{x+3y} \cdot \frac{x+3y}{a-b} = \frac{(a-b)(a+b)}{a-b} = a+b\)

1. н) \(\frac{a^2-b^2}{x+y} \cdot \frac{x-y}{a+b} = \frac{(a-b)(a+b)}{x+y} \cdot \frac{x-y}{a+b} = \frac{(a-b)(x-y)}{x+y}\)

1. о) \(\frac{a^2-b^2}{x+y} : \frac{a+b}{x+y} = \frac{a^2-b^2}{x+y} \cdot \frac{x+y}{a+b} = \frac{(a-b)(a+b)}{a+b} = a-b\)

1. п) \(\frac{x^2+2xy+y^2}{a^2-b^2} = \frac{(x+y)^2}{(a-b)(a+b)}\)

3. а) \(\frac{a^2-9b^2}{c^2+8cd+16d^2} \cdot \frac{c^2-16d^2}{3b-a} = \frac{(a-3b)(a+3b)}{(c+4d)^2} \cdot \frac{(c-4d)(c+4d)}{-(a-3b)} = \frac{(a+3b)(c-4d)}{-(c+4d)} = -\frac{(a+3b)(c-4d)}{c+4d}\)

3. б) \(\frac{a^2-b^2+a+b}{x^2-y^2+x-y} \cdot \frac{3a+3b}{2x-2y} = \frac{(a-b)(a+b)+(a+b)}{(x-y)(x+y)+(x-y)} \cdot \frac{3(a+b)}{2(x-y)} = \frac{(a+b)(a-b+1)}{(x-y)(x+y+1)} \cdot \frac{3(a+b)}{2(x-y)} = \frac{3(a+b)^2(a-b+1)}{2(x-y)^2(x+y+1)}\)

Ответ: 1. а) \(\frac{5}{21}\); б) \(\frac{x^{10}}{18y^{19}}\); в) \(\frac{3a^4b^2}{c^2}\); г) \(\frac{1}{p}\); д) \(\frac{m}{p}\); е) \(\frac{2a}{7b}\); ж) \(\frac{2}{3}\); з) -\(\frac{m-n}{p+q}\); и) \(\frac{a^4b^2}{c^2}\); к) \(\frac{25}{9}\); л) \(\frac{5b}{a-b}\); м) \(a+b\); н) \(\frac{(a-b)(x-y)}{x+y}\); о) \(a-b\); п) \(\frac{(x+y)^2}{(a-b)(a+b)}\). 3. а) -\(\frac{(a+3b)(c-4d)}{c+4d}\); б) \(\frac{3(a+b)^2(a-b+1)}{2(x-y)^2(x+y+1)}\).