Решебник по алгебре 8 класс Мерзляк ФГОС Задание 177

 

\[\boxed{\text{177\ (177).}\text{\ }\text{Еуроки\ -\ ДЗ\ без\ мороки}}\]

\[1)\left( x^{\backslash y} + \frac{x}{y} \right)\ :\left( x^{\backslash y} - \frac{x}{y} \right) =\]

\[= \frac{\text{xy} + x}{y}\ :\frac{\text{xy} - x}{y} =\]

\[= \frac{x(y + 1) \cdot y}{y \cdot x(y - 1)} = \frac{y + 1}{y - 1}\]

\[2)\ \left( \frac{a^{\backslash a - b}}{b} + \frac{a + b^{\backslash b}}{a - b} \right) \cdot \frac{ab^{2}}{a^{2} + b^{2}} =\]

\[= \frac{a^{2} - ab + ab + b^{2}}{b(a - b)} \cdot\]

\[\cdot \frac{ab^{2}}{\left( a^{2} + b^{2} \right)} =\]

\[= \frac{\left( a^{2} + b^{2} \right)ab^{2}}{b(a - b)\left( a^{2} + b^{2} \right)} = \frac{\text{ab}}{a - b}\]

\[3)\ \left( \frac{m}{m - 1} - 1^{\backslash m - 1} \right)\ :\frac{m}{mn - n} =\]

\[= \frac{m - m + 1}{m - 1}\ :\frac{m}{n(m - 1)} =\]

\[= \frac{n(m - 1)}{m(m - 1)} = \frac{n}{m}\]

\[4)\ \left( \frac{a^{\backslash a}}{b} - \frac{b^{\backslash b}}{a} \right) \cdot \frac{4ab}{a - b} =\]

\[= \frac{a^{2} - b^{2}}{\text{ab}} \cdot \frac{4ab}{a - b} =\]

\[= \frac{(a - b) \cdot (a + b) \cdot 4ab}{\text{ab}(a - b)} =\]

\[= 4 \cdot (a + b)\]

\[5)\frac{a}{b} - \frac{a^{2} - b^{2}}{b^{2}}\ :\frac{a + b}{b} =\]

\[= \frac{a}{b} - \frac{(a - b)(a + b) \cdot b}{b^{2}(a + b)} =\]

\[= \frac{a}{b} - \frac{a - b}{b} = \frac{a - a + b}{b} = 1\]

\[6)\ \frac{7x}{x + 2} - \frac{x - 8}{3x + 6} \cdot \frac{84}{x^{2} - 8x} =\]

\[= \frac{7x}{x + 2} - \frac{(x - 8) \cdot 84}{3(x + 2) \cdot x \cdot (x - 8)} =\]

\[= \frac{7x^{\backslash x}}{x + 2} - \frac{28}{x(x + 2)} = \frac{7x^{2} - 28}{x(x + 2)} =\]

\[= \frac{7 \cdot (x - 2)(x + 2)}{x(x + 2)} = \frac{7 \cdot (x - 2)}{x}\]

\[7)\ \left( a^{\backslash a + 3} - \frac{9a - 9}{a + 3} \right)\ :\ \frac{a^{2} - 3a}{a + 3} =\]

\[= \frac{a^{2} + 3a - 9a + 9}{a + 3}\ :\frac{a(a - 3)}{a + 3} =\]

\[= \frac{(a - 3)^{2}(a + 3)}{(a + 3) \cdot a \cdot (a - 3)} = \frac{a - 3}{a}\]

\[8)\ \left( \frac{a^{\backslash a + 8}}{a + 2} - \frac{8^{\backslash a + 2}}{a + 8} \right) \cdot \frac{a^{2} + 8a}{a - 4} =\]

\[= \frac{a^{2} + 8a - 8a - 16}{(a + 2)(a + 8)} \cdot\]

\[\cdot \frac{a(a + 8)}{a - 4} =\]

\[= \frac{(a - 4)(a + 4) \cdot a \cdot (a + 8)}{(a + 2)(a + 8)(a - 4)} =\]

\[= \frac{a(a + 4)}{a + 2}\ \]

\[\boxed{\text{177.}\text{\ }\text{Еуроки\ -\ ДЗ\ без\ мороки}}\]

\[1)\frac{b + 4}{b^{2} - 6b + 9}\ :\frac{b^{2} - 16}{2b - 6} -\]

\[- \frac{2}{b - 4} =\]

\[= \frac{(b + 4) \cdot 2(b - 3)}{(b - 4)(b + 4)(b - 3)^{2}} -\]

\[- \frac{2}{b - 4} =\]

\[= \frac{2}{(b - 4)(b - 3)} - \frac{2^{\backslash b - 3}}{b - 4} =\]

\[= \frac{2 - 2b + 6}{(b - 4)(b - 3)} =\]

\[= \frac{- 2(b - 4)}{(b - 4)(b - 3)} =\]

\[= \frac{- 2}{b - 3} = \frac{2}{3 - b}\]

\[2)\ \left( \frac{m - 1^{\backslash m - 1}}{m + 1} - \frac{m + 1^{\backslash m + 1}}{m - 1} \right)\ :\]

\[:\frac{4m}{m^{2} - 1} =\]

\[= \frac{m^{2} - 2m + 1 - m^{2} - 2m - 1}{(m + 1)(m - 1)} \cdot\]

\[\cdot \frac{m^{2} - 1}{4m} = \frac{- 4m\left( m^{2} - 1 \right)}{\left( m^{2} - 1 \right) \cdot 4m} = - 1\]

\[3)\ \frac{2x}{x^{2} - y^{2}}\ :\]

\[:\left( \frac{1}{x^{2} + 2xy + y^{2}} - \frac{1}{y^{2} - x^{2}} \right) =\]

\[= \frac{2x}{x^{2} - y^{2}}\ :\]

\[:\left( \frac{1^{\backslash x - y}}{(x + y)^{2}} - \frac{1^{\backslash x + y}}{(x + y)(y - x)} \right) =\]

\[= \frac{2x}{x^{2} - y^{2}}\ :\frac{y - x - x - y}{(x + y)^{2}(y - x)} =\]

\[= \frac{2x}{x^{2} - y^{2}} \cdot \frac{{- (x + y)}^{2}(x - y)}{- 2x} =\]

\[= \frac{2x(x + y)^{2}(x - y)}{2x(x + y)(x - y)} = x + y\]

\[4)\ \left( \frac{2a - 3}{a^{2} - 4a + 4} - \frac{a - 1}{a^{2} - 2a} \right)\ :\]

\[:\frac{a^{2} - 2}{a^{3} - 4a} =\]

\[= \left( \frac{2a - 3^{\backslash a}}{(a - 2)^{2}} - \frac{a - 1^{\backslash a - 2}}{a(a - 2)} \right) \cdot\]

\[\cdot \frac{a(a^{2} - 4)}{a^{2} - 2} =\]

\[= \frac{2a^{2} - 3a - a^{2} + 3a - 2}{a(a - 2)^{2}} \cdot\]

\[\cdot \frac{a(a - 2)(a + 2)}{a^{2} - 2} =\]

\[= \frac{\left( a^{2} - 2 \right)a(a - 2)(a + 2)}{a(a - 2)^{2}\left( a^{2} - 2 \right)} =\]

\[= \frac{a + 2}{a - 2}\]