Решебник по алгебре 8 класс Мерзляк Задание 159

\[\boxed{\text{159\ (159).}\text{\ }\text{Еуроки\ -\ ДЗ\ без\ мороки}}\]

\[1)\frac{7a^{2}}{a^{2} - 25} \cdot \frac{5 - a}{a} =\]

\[= \frac{7a^{2}(5 - a)}{(a - 5)(a + 5) \cdot a} = \frac{- 7a}{a + 5}\]

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

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

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

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

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

\[4)\ \frac{a^{2} - 8ab}{12b}\ :\frac{8b^{2} - ab}{24a} =\]

\[= \frac{a(a - 8b) \cdot 24a}{12b \cdot b(8b - a)} = - \frac{2a^{2}}{b^{2}}\]

\[5)\ \frac{5m^{2} - 5n^{2}}{m^{2} + n^{2}}\ :\frac{15n - 15m}{4m^{2} + 4n^{2}} =\]

\[= \frac{5(m - n)(m + n) \cdot 4 \cdot \left( m^{2} + n^{2} \right)}{\left( m^{2} + n^{2} \right) \cdot 15 \cdot (n - m)} =\]

\[= - \frac{4 \cdot (m + n)}{3}\]

\[6)\ \frac{mn^{2} - 36m}{m^{3} - 8}\ :\frac{2n + 12}{6m - 12} =\]

\[= \frac{m(n - 6)(n + 6) \cdot 6 \cdot (m - 2)}{(m - 2)\left( m^{2} + 2m + 4 \right) \cdot 2(n + 6)} =\]

\[= \frac{3m(n - 6)}{m^{2} + 2m + 4}\]

\[7)\ \frac{a^{4} - 1}{a^{2} - a + 1}\ :\frac{a - 1}{a^{3} + 1} =\]

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

\[= \left( a^{2} + 1 \right)(a + 1)^{2}\]

\[8)\ \frac{4x^{2} - 100}{6x}\ :\]

\[:\left( 2x^{2} - 20x + 50 \right) =\]

\[= \frac{4(x - 5)(x + 5)}{6x \cdot 2 \cdot (x - 5)^{2}} = \frac{x + 5}{3x(x - 5)}\]