Решебник по алгебре 8 класс Мерзляк ФГОС Задание 176

 

\[\boxed{\text{176\ (176).}\text{\ }\text{Еуроки\ -\ ДЗ\ без\ мороки}}\]

\[1)\left( \frac{a^{\backslash 4}}{3} + \frac{a^{\backslash 3}}{4} \right) \cdot \frac{6}{a^{2}} = \frac{4a + 3a}{12} \cdot\]

\[\cdot \frac{6}{a^{2}} = \frac{7a \cdot 6}{12a^{2}} = \frac{7}{2a}\]

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

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

\[3)\ \left( 1^{\backslash b} + \frac{a}{b} \right)\ :\left( 1^{\backslash b} - \frac{a}{b} \right) =\]

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

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

\[4)\ \left( \frac{a^{2}}{b^{2}} - \frac{2a^{\backslash b}}{b} + 1^{\backslash b^{2}} \right) \cdot \frac{b}{a - b} =\]

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

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

\[5)\ \frac{a^{2} - ab}{b^{2} - 1} \cdot \frac{b + 1}{a} - \frac{a}{b - 1} =\]

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

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

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

\[6)\ \left( \frac{5^{\backslash m + n}}{m - n} - \frac{4^{\backslash m - n}}{m + n} \right)\ :\frac{m + 9n}{m + n} =\]

\[= \frac{5(m + n) - 4(m - n)}{(m - n)(m + n)}\ :\]

\[:\frac{m + 9n}{m + n} =\]

\[= \frac{(m + 9n)(m + n)}{(m - n)(m + n)(m + 9n)} =\]

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

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

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

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

\[8)\ \frac{x^{2} + x}{4}\ :\frac{x^{2}}{4} + \frac{x - 1}{x} =\]

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

\[= \frac{x + 1 + x - 1}{x} = 2\]

\[9)\ \frac{6c^{2}}{c^{2} - 1}\ :\left( \frac{1}{c - 1} + 1^{\backslash c - 1} \right) =\]

\[= \frac{6c^{2}}{c^{2} - 1}\ :\frac{1 + c - 1}{c - 1} =\]

\[= \frac{6c^{2}(c - 1)}{c(c - 1)(c + 1)} = \frac{6c}{c + 1}\]

\[10)\ \left( \frac{x^{\backslash x - y}}{x + y} + \frac{y^{\backslash x + y}}{x - y} \right) \cdot\]

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

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

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

\[= \frac{x}{x - y}\]

\[\boxed{\text{176.}\text{\ }\text{Еуроки\ -\ ДЗ\ без\ мороки}}\]

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

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

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

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

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

\[= \frac{3 - 3a + 3}{(a - 1)(a - 2)} =\]

\[= \frac{3 - 3(a - 1)}{(a - 1)(a - 2)} =\]

\[= \frac{3(1 - a + 1)}{(a - 1)(a - 2)} =\]

\[= \frac{3(2 - a)}{(a - 1)(a - 2)} =\]

\[= \frac{- 3}{a - 1} = \frac{3}{1 - a}\ \]

\[2)\ \frac{b^{2} + 3b}{b^{3} + 9b} \cdot\]

\[\cdot \left( \frac{b - 3^{\backslash b - 3}}{b + 3} + \frac{b + 3^{\backslash b + 3}}{b - 3} \right) =\]

\[= \frac{b(b + 3)}{b\left( b^{2} + 9 \right)} \cdot\]

\[\cdot \frac{b^{2} - 6b + 9 + b^{2} + 6b + 9}{(b + 3)(b - 3)} =\]

\[= \frac{b(b + 3)\left( 2b^{2} + 18 \right)}{b\left( b^{2} + 9 \right)(b + 3)(b - 3)} =\]

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

\[3)\ \left( \frac{3c + 1^{\backslash 3c + 1}}{3c - 1} - \frac{3c - 1^{\backslash 3c - 1}}{3c + 1} \right)\ :\]

\[:\frac{2c}{6c + 2} =\]

\[= \frac{9c^{2} + 6c + 1 - 9c^{2} + 6c - 1}{(3c - 1)(3c + 1)}\ :\]

\[:\frac{2c}{2(3c + 1)} =\]

\[= \frac{12c}{(3c - 1)(3c + 1)} \cdot \frac{2 \cdot (3c + 1)}{2c} =\]

\[= \frac{12c \cdot 2(3c + 1)}{(3c - 1)(3c + 1) \cdot 2c} = \frac{12}{3c - 1}\]

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

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

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

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

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

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

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

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

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

\[= \frac{1}{a - 2b}\]

\[5)\ \left( \frac{a - 8}{a^{2} - 10a + 25} - \frac{a}{a^{2} - 25} \right)\ :\]

\[:\frac{a - 20}{(a - 5)^{2}} =\]

\[= \left( \frac{a - 8^{\backslash a + 5}}{(a - 5)^{2}} - \frac{a^{\backslash a - 5}}{(a - 5)(a + 5)} \right):\]

\[:\frac{a - 20}{(a - 5)^{2}} =\]

\[= \frac{a^{2} - 3a - 40 - a^{2} + 5a}{(a - 5)^{2}(a + 5)} \cdot\]

\[\cdot \frac{(a - 5)^{2}}{a - 20} =\]

\[= \frac{2(a - 20)(a - 5)^{2}}{(a - 5)^{2}(a + 5)(a - 20)} =\]

\[= \frac{2}{a + 5}\]

\[6)\left( \ \frac{2x + 1}{x^{2} + 6x + 9} - \frac{x - 2}{x^{2} + 3x} \right)\ :\]

\[:\frac{x^{2} + 6}{x^{3} - 9x} =\]

\[= \left( \frac{2x + 1^{\backslash x}}{(x + 3)^{2}} - \frac{x - 2^{\backslash x + 3}}{x(x + 3)} \right):\]

\[:\frac{x^{2} + 6}{x\left( x^{2} - 9 \right)} =\]

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

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

\[= \frac{\left( x^{2} + 6 \right)x(x - 3)(x + 3)}{\left( x^{2} + 6 \right)x(x + 3)^{2}} =\]

\[= \frac{x - 3}{x + 3}\]