Страница 228 номер 783, ГДЗ по алгебре за 9 класс к учебнику Алимова с ответами и решением

1) \(\frac{2 + \sqrt{2} + \sqrt{3}}{\sqrt{2} + \sqrt{3} + \sqrt{6} + \sqrt{8} + 4} = \frac{2 + \sqrt{2} + \sqrt{3}}{\sqrt{2} + \sqrt{3} + \sqrt{2} \cdot \sqrt{3} + 2\sqrt{2} + \sqrt{2} \cdot \sqrt{2} + 2} =\)
\(= \frac{2 + \sqrt{2} + \sqrt{3}}{\sqrt{2}\left(2 + \sqrt{2} + \sqrt{3}\right) + 2 + \sqrt{2} + \sqrt{3}} =\)
\(= \dfrac{2 + \sqrt{2} + \sqrt{3}}{\left(2 + \sqrt{2} + \sqrt{3}\right)\left(\sqrt{2} + 1\right)} = \dfrac{1}{1 + \sqrt{2}} =\)
\(= \dfrac{1 - \sqrt{2}}{\left(1 - \sqrt{2}\right)\left(1 + \sqrt{2}\right)} = \dfrac{1 - \sqrt{2}}{-1} = \sqrt{2} - 1;\)

2) \(\frac{2 - \sqrt{3}}{\sqrt{2} - \sqrt{2 - \sqrt{3}}} + \frac{2 + \sqrt{3}}{\sqrt{2} + \sqrt{2 + \sqrt{3}}} = \frac{\sqrt{2}\left(2 - \sqrt{3}\right)}{2 - \sqrt{4 - 2\sqrt{3}}} + \frac{\sqrt{2}\left(2 + \sqrt{3}\right)}{2 + \sqrt{4 + 2\sqrt{3}}} =\)
\(= \dfrac{\sqrt{2}\left(2 - \sqrt{3}\right)}{2 - \sqrt{3 - 2\sqrt{3} + 1}} + \dfrac{\sqrt{2}\left(2 + \sqrt{3}\right)}{2 + \sqrt{3 + 2\sqrt{3} + 1}} =\)
\(= \dfrac{\sqrt{2}\left(2 - \sqrt{3}\right)}{2 - \sqrt{\left(\sqrt{3} - 1\right)^{2}}} + \dfrac{\sqrt{2}\left(2 + \sqrt{3}\right)}{2 + \sqrt{\left(\sqrt{3} + 1\right)^{2}}} =\)
\(= \dfrac{\sqrt{2}\left(2 - \sqrt{3}\right)}{2 - \sqrt{3} + 1} + \dfrac{\sqrt{2}\left(2 + \sqrt{3}\right)}{2 + \sqrt{3} + 1} =\)
\(= \dfrac{\sqrt{2}\left(2 - \sqrt{3}\right)\left(3 + \sqrt{3}\right) + \sqrt{2}\left(2 + \sqrt{3}\right)\left(3 - \sqrt{3}\right)}{\left(3 - \sqrt{3}\right)\left(3 + \sqrt{3}\right)} =\)
\(= \dfrac{6\sqrt{2} + 2\sqrt{6} - 3\sqrt{6} - 3\sqrt{2} + 6\sqrt{2} - 2\sqrt{6} + 3\sqrt{6} - 3\sqrt{2}}{9 - 3} =\)
\(= \dfrac{6\sqrt{2}}{6} = \sqrt{2};\)

3) \(\frac{\sqrt{x - 2\sqrt{x - 1}} + \sqrt{x + 2\sqrt{x - 1}}}{\sqrt{x^{2} - 4(x - 1)}}\), \(1 < x < 2\)
\(\dfrac{\sqrt{x - 2\sqrt{x - 1}} + \sqrt{x + 2\sqrt{x - 1}}}{\sqrt{x^{2} - 4(x - 1)}} = \dfrac{\sqrt{x - 1 - 2\sqrt{x - 1} + 1} + \sqrt{x - 1 + 2\sqrt{x - 1} + 1}}{\sqrt{x^{2} - 4x + 4}} =\)
\(= \dfrac{\sqrt{\left(\sqrt{x - 1} - 1\right)^{2}} + \sqrt{\left(\sqrt{x - 1} + 1\right)^{2}}}{\sqrt{(x - 2)^{2}}} =\)
\(= \dfrac{\left|\sqrt{x - 1} - 1\right| + \left|\sqrt{x - 1} + 1\right|}{|x - 2|} =\)
\(= \dfrac{1 - \sqrt{x - 1} + \sqrt{x - 1} + 1}{2 - x} = \dfrac{2}{2 - x};\)

4) \(\left(\frac{a + b}{\sqrt[3]{a^{2}} - \sqrt[3]{b^{2}}} + \frac{\sqrt[3]{ab^{2}} - \sqrt[3]{a^{2}b}}{\sqrt[3]{a^{2}} - 2\sqrt[3]{ab} + \sqrt[3]{b^{2}}}\right) : \left(\sqrt[6]{a} - \sqrt[6]{b}\right) - \sqrt[6]{b} =\)
\(= \left(\dfrac{a + b}{\left(\sqrt[3]{a} - \sqrt[3]{b}\right)\left(\sqrt[3]{a} + \sqrt[3]{b}\right)} + \dfrac{\sqrt[3]{ab}\left(\sqrt[3]{b} - \sqrt[3]{a}\right)}{\left(\sqrt[3]{a} - \sqrt[3]{b}\right)^{2}}\right) :\)
\(: \left(\sqrt[6]{a} - \sqrt[6]{b}\right) - \sqrt[6]{b} =\)
\(= \left(\dfrac{a + b - \sqrt[3]{ab}\left(\sqrt[3]{a} + \sqrt[3]{b}\right)}{\left(\sqrt[3]{a} - \sqrt[3]{b}\right)\left(\sqrt[3]{a} + \sqrt[3]{b}\right)}\right) : \left(\sqrt[6]{a} - \sqrt[6]{b}\right) - \sqrt[6]{b} =\)
\(= \left(\dfrac{\left(\sqrt[3]{a} + \sqrt[3]{b}\right)\left(\sqrt[3]{a^{2}} - \sqrt[3]{ab} + \sqrt[3]{b^{2}}\right) - \sqrt[3]{ab}\left(\sqrt[3]{a} + \sqrt[3]{b}\right)}{\left(\sqrt[3]{a} - \sqrt[3]{b}\right)\left(\sqrt[3]{a} + \sqrt[3]{b}\right)}\right) :\)
\(: \left(\sqrt[6]{a} - \sqrt[6]{b}\right) - \sqrt[6]{b} =\)
\(= \left(\dfrac{\left(\sqrt[3]{a} + \sqrt[3]{b}\right)\left(\sqrt[3]{a} - \sqrt[3]{b}\right)^{2}}{\left(\sqrt[3]{a} - \sqrt[3]{b}\right)\left(\sqrt[3]{a} + \sqrt[3]{b}\right)}\right) : \left(\sqrt[6]{a} - \sqrt[6]{b}\right) - \sqrt[6]{b} =\)
\(= \dfrac{\left(\sqrt[6]{a} - \sqrt[6]{b}\right)\left(\sqrt[6]{a} + \sqrt[6]{b}\right)}{\sqrt[6]{a} - \sqrt[6]{b}} - \sqrt[6]{b} = \sqrt[6]{a}.\)