Решебник по алгебре 8 класс Мерзляк ФГОС Задание 164

\[\boxed{\text{164\ (164).}\text{\ }\text{Еуроки\ -\ ДЗ\ без\ мороки}}\]

\[x^{2} + \frac{16}{x^{2}} = 41;\ \ \ \ \ \ x + \frac{4}{x} = ?\]

\[\left( x + \frac{4}{x} \right)^{2} = x^{2} + 8 + \frac{16}{x^{2}}\]

\[\left( x + \frac{4}{x} \right)^{2} = 41 + 8\]

\[\left( x + \frac{4}{x} \right)^{2} = 49\]

\[\left\{ \begin{matrix} x + \frac{4}{x} = 7\ \ \ \ \\ x + \frac{4}{x} = - 7 \\ \end{matrix} \right.\ \]

\[Ответ:7;\ - 7.\]

\[\boxed{\text{164.}\text{\ }\text{Еуроки\ -\ ДЗ\ без\ мороки}}\]

\[1)\frac{a^{2} - 36}{a^{2} + ab - 6a - 6b}\ :\]

\[:\frac{a^{2} + ab + 6a + 6b}{a^{2} + 2ab + b^{2}} =\]

\[= \frac{(a - 6)(a + 6)(a + b)^{2}}{\left( a^{2} + ab - 6a - 6b \right)\left( a^{2} + ab + 6a + 6b \right)} =\]

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

\[= \frac{(a - 6)(a + 6)(a + b)^{2}}{(a + b)(a - 6)(a + b)(a + 6)} =\]

\[= 1\]

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

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

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

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

\[= 1\]