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МатематикаРешебник по алгебре 8 класс Мерзляк ФГОС Задание 769
Задание 769
\[\boxed{\mathbf{769\ (769).\ }Еуроки\ - \ ДЗ\ без\ мороки}\]
\[1)\ \frac{3 + \sqrt{3}}{2\sqrt{3}} = \frac{\sqrt{3} \cdot \left( \sqrt{3} + 1 \right)}{2\sqrt{3}} =\]
\[= \frac{\sqrt{3} + 1}{2}\]
\[2)\ \frac{5 - \sqrt{5}}{\sqrt{10} - 5\sqrt{2}} = \frac{5 - \sqrt{5}}{\sqrt{2}\left( \sqrt{5} - 5 \right)} =\]
\[= - \frac{1}{\sqrt{2}}\]
\[3)\ \frac{2 - \sqrt{6}}{\sqrt{6} - 3\ } = \frac{\sqrt{2} \cdot \left( \sqrt{2} - \sqrt{3} \right)}{\sqrt{3} \cdot \left( \sqrt{2} - \sqrt{3} \right)} =\]
\[= \frac{\sqrt{2}}{\sqrt{3}} = \sqrt{\frac{2}{3}}\ \]
\[4)\ \frac{4a - 2}{2\sqrt{a} + \sqrt{2}} =\]
\[= \frac{\left( 2\sqrt{a} - \sqrt{2} \right)\left( 2\sqrt{a} + \sqrt{2} \right)}{\left( 2\sqrt{a} + \sqrt{2} \right)} =\]
\[= 2\sqrt{a} - \sqrt{2}\]
\[5)\ \frac{9a - b^{2}}{9a + 6b\sqrt{a} + b^{2}} =\]
\[= \frac{\left( 3\sqrt{a} - b \right)\left( 3\sqrt{a} + b \right)}{\left( 3\sqrt{a} + b \right)^{2}} =\]
\[= \frac{3\sqrt{a} - b}{3\sqrt{a} + b}\]
\[6)\ \frac{a\sqrt{a} - 8}{a + 2\sqrt{a} + 4} =\]
\[= \frac{\left( \sqrt{a} - 2 \right)\left( a + 2\sqrt{a} + 4 \right)}{\left( a + 2\sqrt{a} + 4 \right)} =\]
\[= \sqrt{a} - 2\]
\[\boxed{\mathbf{7}\mathbf{6}\mathbf{9}\mathbf{\text{.\ }}Еуроки\ - \ ДЗ\ без\ мороки}\]
\[x^{2} + 5x - 16 = 0\]
\[x_{1} + x_{2} = - 5;\ \ \ \ x_{1} \cdot x_{2} = - 16\]
\[1)\text{\ x}_{1}²x_{2} + x_{1}x_{2}² =\]
\[= x_{1}\left( x_{1}x_{2} + x_{2}^{2} \right) =\]
\[= x_{1}x_{2}\left( x_{1}{+ x}_{2} \right) = - 16 \cdot ( - 5) =\]
\[= 80\]
\[2)\ \frac{x_{2}}{x_{1}} + \frac{x_{1}}{x_{2}} = \frac{x_{2}^{2} + x_{1}^{2}}{x_{1}x_{2}} =\]
\[= \frac{x_{2}^{2} + x_{1}^{2} + 2x_{1}x_{2} - 2x_{1}x_{2}}{x_{1}x_{2}} =\]
\[= \frac{\left( x_{1} + x_{2} \right)^{2} - 2x_{1}x_{2}}{x_{1}x_{2}} =\]
\[= \frac{25 + 32}{- 16} = - \frac{57}{16} = - 3\frac{9}{16}\]
\[3)\ \left| x_{2} - x_{1} \right| = \sqrt{\left( x_{2} - x_{1} \right)^{2}} =\]
\[= \sqrt{\left( x_{2}^{2} + x_{1}^{2} - 2x_{1}x_{2} \right)} =\]
\[= \sqrt{\left( x_{1} + x_{2} \right)^{2} - 4x_{1}x_{2}} =\]
\[= \sqrt{25 - 4 \cdot ( - 16)} = \sqrt{89}\ \]
