Найдите значение выражения 1. \(\sqrt{\frac{5}{\sqrt{6}-1}}-\sqrt{6}\) 2. \(\sqrt{\frac{4-8\sqrt{5}}{1-\sqrt{5}}}-\sqrt{5}\) 3. \(\sqrt{\frac{30-5\sqrt{6}}{4-\sqrt{6}}}-\sqrt{6}\) 4. \(\sqrt{4\sqrt{5}+9}-\sqrt{5}\) 5. \(\frac{6}{3+\sqrt{7}}+3\sqrt{7}\) 6. \(\sqrt{6\sqrt{5}+14}-\sqrt{5}\) 7. \(\sqrt{2\sqrt{5}+6}-\sqrt{5}\) 8. \(\frac{52}{4+\sqrt{3}}+4\sqrt{3}\) 9. \(\frac{2}{2+\sqrt{3}}+2\sqrt{3}\) 10. \(\sqrt{\frac{2}{\sqrt{5}-2}}-2\sqrt{5}\)

Решение заданий:

1. \(\sqrt{\frac{5}{\sqrt{6}-1}}-\sqrt{6} = \sqrt{\frac{5(\sqrt{6}+1)}{(\sqrt{6}-1)(\sqrt{6}+1)}}-\sqrt{6} = \sqrt{\frac{5(\sqrt{6}+1)}{6-1}}-\sqrt{6} = \sqrt{\sqrt{6}+1}-\sqrt{6}\) 2. \(\sqrt{\frac{4-8\sqrt{5}}{1-\sqrt{5}}}-\sqrt{5} = \sqrt{\frac{(4-8\sqrt{5})(1+\sqrt{5})}{(1-\sqrt{5})(1+\sqrt{5})}}-\sqrt{5} = \sqrt{\frac{4+4\sqrt{5}-8\sqrt{5}-40}{1-5}}-\sqrt{5} = \sqrt{\frac{-36-4\sqrt{5}}{-4}}-\sqrt{5} = \sqrt{9+\sqrt{5}}-\sqrt{5}\) 3. \(\sqrt{\frac{30-5\sqrt{6}}{4-\sqrt{6}}}-\sqrt{6} = \sqrt{\frac{(30-5\sqrt{6})(4+\sqrt{6})}{(4-\sqrt{6})(4+\sqrt{6})}}-\sqrt{6} = \sqrt{\frac{120+30\sqrt{6}-20\sqrt{6}-30}{16-6}}-\sqrt{6} = \sqrt{\frac{90+10\sqrt{6}}{10}}-\sqrt{6} = \sqrt{9+\sqrt{6}}-\sqrt{6}\) 4. \(\sqrt{4\sqrt{5}+9}-\sqrt{5}\) 5. \(\frac{6}{3+\sqrt{7}}+3\sqrt{7} = \frac{6(3-\sqrt{7})}{(3+\sqrt{7})(3-\sqrt{7})}+3\sqrt{7} = \frac{18-6\sqrt{7}}{9-7}+3\sqrt{7} = \frac{18-6\sqrt{7}}{2}+3\sqrt{7} = 9-3\sqrt{7}+3\sqrt{7} = 9\) 6. \(\sqrt{6\sqrt{5}+14}-\sqrt{5}\) 7. \(\sqrt{2\sqrt{5}+6}-\sqrt{5}\) 8. \(\frac{52}{4+\sqrt{3}}+4\sqrt{3} = \frac{52(4-\sqrt{3})}{(4+\sqrt{3})(4-\sqrt{3})}+4\sqrt{3} = \frac{52(4-\sqrt{3})}{16-3}+4\sqrt{3} = \frac{52(4-\sqrt{3})}{13}+4\sqrt{3} = 4(4-\sqrt{3})+4\sqrt{3} = 16-4\sqrt{3}+4\sqrt{3} = 16\) 9. \(\frac{2}{2+\sqrt{3}}+2\sqrt{3} = \frac{2(2-\sqrt{3})}{(2+\sqrt{3})(2-\sqrt{3})}+2\sqrt{3} = \frac{4-2\sqrt{3}}{4-3}+2\sqrt{3} = 4-2\sqrt{3}+2\sqrt{3} = 4\) 10. \(\sqrt{\frac{2}{\sqrt{5}-2}}-2\sqrt{5} = \sqrt{\frac{2(\sqrt{5}+2)}{(\sqrt{5}-2)(\sqrt{5}+2)}}-2\sqrt{5} = \sqrt{\frac{2(\sqrt{5}+2)}{5-4}}-2\sqrt{5} = \sqrt{2(\sqrt{5}+2)}-2\sqrt{5} = \sqrt{2\sqrt{5}+4}-2\sqrt{5}\)

Ответ: [смотрите выше]

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