Страница 41 номер 2.3.14, ГДЗ по алгебре за 7, 8 и 9 класс к задачнику Ткачевой
Выполнить действия: 1) \(\frac{x + y}{3x^3 y^5} \cdot \frac{x^2 - y^2}{5x^2 y} : \frac{(x + y)^2}{60x^6 y^6}\); 2) \(\frac{a + b}{2b - a} : \frac{(a + b)^2}{a^2 - 4b^2} \cdot \frac{6a^2 b}{2b + a}\); 3) \(\frac{3a - 3b}{2ab} \cdot \left(\frac{1}{a - b} - \frac{1}{a + b}\right)\); 4) \(\frac{2x + 2y}{6xy} \cdot \left(\frac{3}{x - y} - \frac{3}{x + y}\right)\); 5) \(\left(\frac{2}{m - n} - \frac{2}{m + n}\right) : \frac{7}{m + n}\); 6) \(\left(\frac{1}{p - q} - \frac{1}{p + q}\right) : \frac{5}{6p - 6q}\); 7) \(\left(\frac{1}{x} + \frac{1}{y}\right) \cdot \frac{3xy}{x^2 - y^2}\); 8) \(\left(\frac{1}{x} - \frac{1}{y}\right) \cdot \frac{xy}{x^2 - y^2}\); 9) \(\left(\frac{b}{a} - \frac{a}{b}\right) : \left(\frac{1}{b} - \frac{1}{a}\right)\); 10) \(\left(a + \frac{a}{b}\right) : \left(a - \frac{a}{b}\right)\); 11) \(\left(\frac{2m}{m - n} + \frac{m - n}{n}\right) \cdot n\); 12) \(\left(\frac{m + n}{m} - \frac{2n}{m + n}\right) \cdot (m + n)\); 13) \(\frac{3}{x} - \frac{x^2 - 9}{x^3} \cdot \frac{x}{x + 3}\); 14) \(\frac{5}{y} - \frac{y}{x - y} \cdot \frac{x^2 - y^2}{y^2}\); 15) \(\frac{y}{x - y} : \left(\frac{x}{x - y} - \frac{x + y}{x}\right)\); 16) \(\frac{a}{a + b} : \left(\frac{b}{a + b} + \frac{a - b}{b}\right)\).
1) \[\begin{aligned} &\frac{x + y}{3x^3 y^5} \cdot \frac{x^2 - y^2}{5x^2 y} : \frac{(x + y)^2}{60x^6 y^6} = {} \\ &= \frac{x + y}{3x^3 y^5} \cdot \frac{(x - y)(x + y)}{5x^2 y} \cdot \frac{60x^6 y^6}{(x + y)^2} = {} \\ &= \frac{60x^6 y^6 (x + y)^2 (x - y)}{15x^5 y^6 (x + y)^2} = 4x(x - y). \end{aligned}\]
2) \[\begin{aligned} &\frac{a + b}{2b - a} : \frac{(a + b)^2}{a^2 - 4b^2} \cdot \frac{6a^2 b}{2b + a} = {} \\ &= \frac{a + b}{-(a - 2b)} \cdot \frac{(a - 2b)(a + 2b)}{(a + b)^2} \cdot \frac{6a^2 b}{a + 2b} = {} \\ &= -\frac{6a^2 b}{a + b}. \end{aligned}\]
3) \[\begin{aligned} &\frac{3a - 3b}{2ab} \cdot \left(\frac{1}{a - b} - \frac{1}{a + b}\right) = \frac{3(a - b)}{2ab} \cdot \frac{(a + b) - (a - b)}{(a - b)(a + b)} = {} \\ &= \frac{3(a - b) \cdot 2b}{2ab(a - b)(a + b)} = \frac{3}{a(a + b)}. \end{aligned}\]
4) \[\begin{aligned} &\frac{2x + 2y}{6xy} \cdot \left(\frac{3}{x - y} - \frac{3}{x + y}\right) = \frac{2(x + y)}{6xy} \cdot \frac{3(x + y) - 3(x - y)}{(x - y)(x + y)} = {} \\ &= \frac{2(x + y) \cdot 6y}{6xy(x - y)(x + y)} = \frac{2}{x(x - y)}. \end{aligned}\]
5) \[\begin{aligned} &\left(\frac{2}{m - n} - \frac{2}{m + n}\right) : \frac{7}{m + n} = \frac{2(m + n) - 2(m - n)}{(m - n)(m + n)} \cdot \frac{m + n}{7} = {} \\ &= \frac{4n(m + n)}{7(m - n)(m + n)} = \frac{4n}{7(m - n)}. \end{aligned}\]
6) \[\begin{aligned} &\left(\frac{1}{p - q} - \frac{1}{p + q}\right) : \frac{5}{6p - 6q} = \frac{(p + q) - (p - q)}{(p - q)(p + q)} \cdot \frac{6(p - q)}{5} = {} \\ &= \frac{2q \cdot 6(p - q)}{5(p - q)(p + q)} = \frac{12q}{5(p + q)}. \end{aligned}\]
7) \[\begin{aligned} &\left(\frac{1}{x} + \frac{1}{y}\right) \cdot \frac{3xy}{x^2 - y^2} = \frac{y + x}{xy} \cdot \frac{3xy}{(x - y)(x + y)} = {} \\ &= \frac{3}{x - y}. \end{aligned}\]
8) \[\begin{aligned} &\left(\frac{1}{x} - \frac{1}{y}\right) \cdot \frac{xy}{x^2 - y^2} = \frac{y - x}{xy} \cdot \frac{xy}{(x - y)(x + y)} = {} \\ &= \frac{-(x - y)}{(x - y)(x + y)} = -\frac{1}{x + y}. \end{aligned}\]
9) \[\begin{aligned} &\left(\frac{b}{a} - \frac{a}{b}\right) : \left(\frac{1}{b} - \frac{1}{a}\right) = \frac{b^2 - a^2}{ab} \cdot \frac{ab}{a - b} = {} \\ &= \frac{(b - a)(b + a)}{a - b} = \frac{-(a - b)(a + b)}{a - b} = -(a + b). \end{aligned}\]
10) \[\begin{aligned} &\left(a + \frac{a}{b}\right) : \left(a - \frac{a}{b}\right) = \frac{ab + a}{b} \cdot \frac{b}{ab - a} = {} \\ &= \frac{a(b + 1)}{a(b - 1)} = \frac{b + 1}{b - 1}. \end{aligned}\]
11) \[\begin{aligned} &\left(\frac{2m}{m - n} + \frac{m - n}{n}\right) \cdot n = \frac{2mn + (m - n)^2}{n(m - n)} \cdot n = {} \\ &= \frac{2mn + m^2 - 2mn + n^2}{m - n} = \frac{m^2 + n^2}{m - n}. \end{aligned}\]
12) \[\begin{aligned} &\left(\frac{m + n}{m} - \frac{2n}{m + n}\right) \cdot (m + n) = \frac{(m + n)^2 - 2mn}{m(m + n)} \cdot (m + n) = {} \\ &= \frac{m^2 + 2mn + n^2 - 2mn}{m} = \frac{m^2 + n^2}{m}. \end{aligned}\]
13) \[\begin{aligned} &\frac{3}{x} - \frac{x^2 - 9}{x^3} \cdot \frac{x}{x + 3} = \frac{3}{x} - \frac{(x - 3)(x + 3) \cdot x}{x^3 (x + 3)} = {} \\ &= \frac{3}{x} - \frac{x - 3}{x^2} = \frac{3x - (x - 3)}{x^2} = \frac{2x + 3}{x^2}. \end{aligned}\]
14) \[\begin{aligned} &\frac{5}{y} - \frac{y}{x - y} \cdot \frac{x^2 - y^2}{y^2} = \frac{5}{y} - \frac{y(x - y)(x + y)}{(x - y)y^2} = {} \\ &= \frac{5}{y} - \frac{x + y}{y} = \frac{5 - x - y}{y}. \end{aligned}\]
15) \[\begin{aligned} &\frac{y}{x - y} : \left(\frac{x}{x - y} - \frac{x + y}{x}\right) = \frac{y}{x - y} : \frac{x^2 - (x + y)(x - y)}{x(x - y)} = {} \\ &= \frac{y}{x - y} : \frac{y^2}{x(x - y)} = \frac{y}{x - y} \cdot \frac{x(x - y)}{y^2} = \frac{x}{y}. \end{aligned}\]
16) \[\begin{aligned} &\frac{a}{a + b} : \left(\frac{b}{a + b} + \frac{a - b}{b}\right) = \frac{a}{a + b} : \frac{b^2 + (a - b)(a + b)}{b(a + b)} = {} \\ &= \frac{a}{a + b} : \frac{a^2}{b(a + b)} = \frac{a}{a + b} \cdot \frac{b(a + b)}{a^2} = \frac{b}{a}. \end{aligned}\]
Ответ: 1) \(4x(x - y)\); 2) \(-\frac{6a^2 b}{a + b}\); 3) \(\frac{3}{a(a + b)}\); 4) \(\frac{2}{x(x - y)}\); 5) \(\frac{4n}{7(m - n)}\); 6) \(\frac{12q}{5(p + q)}\); 7) \(\frac{3}{x - y}\); 8) \(-\frac{1}{x + y}\); 9) \(-(a + b)\); 10) \(\frac{b + 1}{b - 1}\); 11) \(\frac{m^2 + n^2}{m - n}\); 12) \(\frac{m^2 + n^2}{m}\); 13) \(\frac{2x + 3}{x^2}\); 14) \(\frac{5 - x - y}{y}\); 15) \(\frac{x}{y}\); 16) \(\frac{b}{a}\).
Выполнить действия: 1) \(\frac{x + y}{3x^3 y^5} \cdot \frac{x^2 - y^2}{5x^2 y} : \frac{(x + y)^2}{60x^6 y^6}\); 2) \(\frac{a + b}{2b - a} : \frac{(a + b)^2}{a^2 - 4b^2} \cdot \frac{6a^2 b}{2b + a}\); 3) \(\frac{3a - 3b}{2ab} \cdot \left(\frac{1}{a - b} - \frac{1}{a + b}\right)\); 4) \(\frac{2x + 2y}{6xy} \cdot \left(\frac{3}{x - y} - \frac{3}{x + y}\right)\); 5) \(\left(\frac{2}{m - n} - \frac{2}{m + n}\right) : \frac{7}{m + n}\); 6) \(\left(\frac{1}{p - q} - \frac{1}{p + q}\right) : \frac{5}{6p - 6q}\); 7) \(\left(\frac{1}{x} + \frac{1}{y}\right) \cdot \frac{3xy}{x^2 - y^2}\); 8) \(\left(\frac{1}{x} - \frac{1}{y}\right) \cdot \frac{xy}{x^2 - y^2}\); 9) \(\left(\frac{b}{a} - \frac{a}{b}\right) : \left(\frac{1}{b} - \frac{1}{a}\right)\); 10) \(\left(a + \frac{a}{b}\right) : \left(a - \frac{a}{b}\right)\); 11) \(\left(\frac{2m}{m - n} + \frac{m - n}{n}\right) \cdot n\); 12) \(\left(\frac{m + n}{m} - \frac{2n}{m + n}\right) \cdot (m + n)\); 13) \(\frac{3}{x} - \frac{x^2 - 9}{x^3} \cdot \frac{x}{x + 3}\); 14) \(\frac{5}{y} - \frac{y}{x - y} \cdot \frac{x^2 - y^2}{y^2}\); 15) \(\frac{y}{x - y} : \left(\frac{x}{x - y} - \frac{x + y}{x}\right)\); 16) \(\frac{a}{a + b} : \left(\frac{b}{a + b} + \frac{a - b}{b}\right)\).
Действие в скобках выполняем приведением дробей к общему знаменателю, деление заменяем умножением на дробь, обратную делителю, а полученную дробь сокращаем после разложения её числителя и знаменателя на множители.
1) \[\begin{aligned} &\frac{x + y}{3x^3 y^5} \cdot \frac{x^2 - y^2}{5x^2 y} : \frac{(x + y)^2}{60x^6 y^6} = {} \\ &= \frac{x + y}{3x^3 y^5} \cdot \frac{(x - y)(x + y)}{5x^2 y} \cdot \frac{60x^6 y^6}{(x + y)^2} = {} \\ &= \frac{60x^6 y^6 (x + y)^2 (x - y)}{15x^5 y^6 (x + y)^2} = 4x(x - y). \end{aligned}\]
2) \[\begin{aligned} &\frac{a + b}{2b - a} : \frac{(a + b)^2}{a^2 - 4b^2} \cdot \frac{6a^2 b}{2b + a} = {} \\ &= \frac{a + b}{-(a - 2b)} \cdot \frac{(a - 2b)(a + 2b)}{(a + b)^2} \cdot \frac{6a^2 b}{a + 2b} = {} \\ &= -\frac{6a^2 b}{a + b}. \end{aligned}\]
3) \[\begin{aligned} &\frac{3a - 3b}{2ab} \cdot \left(\frac{1}{a - b} - \frac{1}{a + b}\right) = \frac{3(a - b)}{2ab} \cdot \frac{(a + b) - (a - b)}{(a - b)(a + b)} = {} \\ &= \frac{3(a - b) \cdot 2b}{2ab(a - b)(a + b)} = \frac{3}{a(a + b)}. \end{aligned}\]
4) \[\begin{aligned} &\frac{2x + 2y}{6xy} \cdot \left(\frac{3}{x - y} - \frac{3}{x + y}\right) = \frac{2(x + y)}{6xy} \cdot \frac{3(x + y) - 3(x - y)}{(x - y)(x + y)} = {} \\ &= \frac{2(x + y) \cdot 6y}{6xy(x - y)(x + y)} = \frac{2}{x(x - y)}. \end{aligned}\]
5) \[\begin{aligned} &\left(\frac{2}{m - n} - \frac{2}{m + n}\right) : \frac{7}{m + n} = \frac{2(m + n) - 2(m - n)}{(m - n)(m + n)} \cdot \frac{m + n}{7} = {} \\ &= \frac{4n(m + n)}{7(m - n)(m + n)} = \frac{4n}{7(m - n)}. \end{aligned}\]
6) \[\begin{aligned} &\left(\frac{1}{p - q} - \frac{1}{p + q}\right) : \frac{5}{6p - 6q} = \frac{(p + q) - (p - q)}{(p - q)(p + q)} \cdot \frac{6(p - q)}{5} = {} \\ &= \frac{2q \cdot 6(p - q)}{5(p - q)(p + q)} = \frac{12q}{5(p + q)}. \end{aligned}\]
7) \[\begin{aligned} &\left(\frac{1}{x} + \frac{1}{y}\right) \cdot \frac{3xy}{x^2 - y^2} = \frac{y + x}{xy} \cdot \frac{3xy}{(x - y)(x + y)} = {} \\ &= \frac{3}{x - y}. \end{aligned}\]
8) \[\begin{aligned} &\left(\frac{1}{x} - \frac{1}{y}\right) \cdot \frac{xy}{x^2 - y^2} = \frac{y - x}{xy} \cdot \frac{xy}{(x - y)(x + y)} = {} \\ &= \frac{-(x - y)}{(x - y)(x + y)} = -\frac{1}{x + y}. \end{aligned}\]
9) \[\begin{aligned} &\left(\frac{b}{a} - \frac{a}{b}\right) : \left(\frac{1}{b} - \frac{1}{a}\right) = \frac{b^2 - a^2}{ab} \cdot \frac{ab}{a - b} = {} \\ &= \frac{(b - a)(b + a)}{a - b} = \frac{-(a - b)(a + b)}{a - b} = -(a + b). \end{aligned}\]
10) \[\begin{aligned} &\left(a + \frac{a}{b}\right) : \left(a - \frac{a}{b}\right) = \frac{ab + a}{b} \cdot \frac{b}{ab - a} = {} \\ &= \frac{a(b + 1)}{a(b - 1)} = \frac{b + 1}{b - 1}. \end{aligned}\]
11) \[\begin{aligned} &\left(\frac{2m}{m - n} + \frac{m - n}{n}\right) \cdot n = \frac{2mn + (m - n)^2}{n(m - n)} \cdot n = {} \\ &= \frac{2mn + m^2 - 2mn + n^2}{m - n} = \frac{m^2 + n^2}{m - n}. \end{aligned}\]
12) \[\begin{aligned} &\left(\frac{m + n}{m} - \frac{2n}{m + n}\right) \cdot (m + n) = \frac{(m + n)^2 - 2mn}{m(m + n)} \cdot (m + n) = {} \\ &= \frac{m^2 + 2mn + n^2 - 2mn}{m} = \frac{m^2 + n^2}{m}. \end{aligned}\]
13) \[\begin{aligned} &\frac{3}{x} - \frac{x^2 - 9}{x^3} \cdot \frac{x}{x + 3} = \frac{3}{x} - \frac{(x - 3)(x + 3) \cdot x}{x^3 (x + 3)} = {} \\ &= \frac{3}{x} - \frac{x - 3}{x^2} = \frac{3x - (x - 3)}{x^2} = \frac{2x + 3}{x^2}. \end{aligned}\]
14) \[\begin{aligned} &\frac{5}{y} - \frac{y}{x - y} \cdot \frac{x^2 - y^2}{y^2} = \frac{5}{y} - \frac{y(x - y)(x + y)}{(x - y)y^2} = {} \\ &= \frac{5}{y} - \frac{x + y}{y} = \frac{5 - x - y}{y}. \end{aligned}\]
15) \[\begin{aligned} &\frac{y}{x - y} : \left(\frac{x}{x - y} - \frac{x + y}{x}\right) = \frac{y}{x - y} : \frac{x^2 - (x + y)(x - y)}{x(x - y)} = {} \\ &= \frac{y}{x - y} : \frac{y^2}{x(x - y)} = \frac{y}{x - y} \cdot \frac{x(x - y)}{y^2} = \frac{x}{y}. \end{aligned}\]
16) \[\begin{aligned} &\frac{a}{a + b} : \left(\frac{b}{a + b} + \frac{a - b}{b}\right) = \frac{a}{a + b} : \frac{b^2 + (a - b)(a + b)}{b(a + b)} = {} \\ &= \frac{a}{a + b} : \frac{a^2}{b(a + b)} = \frac{a}{a + b} \cdot \frac{b(a + b)}{a^2} = \frac{b}{a}. \end{aligned}\]
Ответ: 1) \(4x(x - y)\); 2) \(-\frac{6a^2 b}{a + b}\); 3) \(\frac{3}{a(a + b)}\); 4) \(\frac{2}{x(x - y)}\); 5) \(\frac{4n}{7(m - n)}\); 6) \(\frac{12q}{5(p + q)}\); 7) \(\frac{3}{x - y}\); 8) \(-\frac{1}{x + y}\); 9) \(-(a + b)\); 10) \(\frac{b + 1}{b - 1}\); 11) \(\frac{m^2 + n^2}{m - n}\); 12) \(\frac{m^2 + n^2}{m}\); 13) \(\frac{2x + 3}{x^2}\); 14) \(\frac{5 - x - y}{y}\); 15) \(\frac{x}{y}\); 16) \(\frac{b}{a}\).
