Контрольные задания > Please solve the following math problems: 1. \(\sqrt{\frac{3 \cdot 42}{70}} = \sqrt{0} = 3\) 2. \(\frac{\sqrt{15} \cdot \sqrt{12}}{\sqrt{20}} = \sqrt{\frac{15 \cdot 12}{20}} = \sqrt{\frac{180}{20}} = \sqrt{9} = 3\) 3. \(\frac{\sqrt{21} \cdot \sqrt{14}}{\sqrt{6}} = \sqrt{\frac{21 \cdot 14}{6}} = \sqrt{\frac{294}{6}} = \sqrt{49} = 7\) 4. \((\sqrt{27} - \sqrt{3}) \cdot \sqrt{3} = \sqrt{27 \cdot 3} - \sqrt{3 \cdot 3} = \sqrt{81} - \sqrt{9} = 9 - 3 = 6\) 5. \((\sqrt{17} - 3) \cdot (\sqrt{17} + 3) = (\sqrt{17})^2 - 3^2 = 17 - 9 = 8\) 6. \(\frac{5 \sqrt{7} \cdot 2 \sqrt{2} \cdot \sqrt{22}}{?} = 10 \sqrt{7 \cdot 2 \cdot 22} = 10 \sqrt{308}\) 7. \(2\sqrt{13} \cdot 5\sqrt{2} \cdot \sqrt{26} = 10 \sqrt{13 \cdot 2 \cdot 26} = 10 \sqrt{676} = 10 \cdot 26 = 260\) 8. \((\sqrt{12} + \sqrt{3}) \cdot \sqrt{3} = \sqrt{12 \cdot 3} + \sqrt{3 \cdot 3} = \sqrt{36} + \sqrt{9} = 6 + 3 = 9\) 9. \((\sqrt{23} - 2) \cdot (\sqrt{23} + 2) = (\sqrt{23})^2 - 2^2 = 23 - 4 = 19\) 10. \(\frac{\sqrt{32} \cdot \sqrt{6}}{\sqrt{12}} = \sqrt{\frac{32 \cdot 6}{12}} = \sqrt{\frac{192}{12}} = \sqrt{16} = 4\) 11. \(7\sqrt{15} \cdot 2\sqrt{2} \cdot \sqrt{30} = 14\sqrt{15 \cdot 2 \cdot 30} = 14\sqrt{900} = 14 \cdot 30 = 420\) 12. \(\frac{\sqrt{35} \cdot \sqrt{21}}{\sqrt{15}} = \sqrt{\frac{35 \cdot 21}{15}} = \sqrt{\frac{735}{15}} = \sqrt{49} = 7\) (Letter A) 13. \(4\sqrt{17} \cdot 5\sqrt{2} \cdot \sqrt{34} = 20\sqrt{17 \cdot 2 \cdot 34} = 20\sqrt{1156} = 20 \cdot 34 = 680\) 14. \(\frac{\sqrt{20} \cdot \sqrt{32}}{\sqrt{10}} = \sqrt{\frac{20 \cdot 32}{10}} = \sqrt{\frac{640}{10}} = \sqrt{64} = 8\) (Letter C) 15. \(5\sqrt{7} \cdot 6\sqrt{3} \cdot \sqrt{21} = 30\sqrt{7 \cdot 3 \cdot 21} = 30\sqrt{441} = 30 \cdot 21 = 630\) 16. \((\sqrt{18} - \sqrt{2}) \cdot \sqrt{2} = \sqrt{18 \cdot 2} - \sqrt{2 \cdot 2} = \sqrt{36} - \sqrt{4} = 6 - 2 = 4\) 17. \((\sqrt{11} - 3) \cdot (\sqrt{11} + 3) = (\sqrt{11})^2 - 3^2 = 11 - 9 = 2\)
Вопрос:

Please solve the following math problems: 1. \(\sqrt{\frac{3 \cdot 42}{70}} = \sqrt{0} = 3\) 2. \(\frac{\sqrt{15} \cdot \sqrt{12}}{\sqrt{20}} = \sqrt{\frac{15 \cdot 12}{20}} = \sqrt{\frac{180}{20}} = \sqrt{9} = 3\) 3. \(\frac{\sqrt{21} \cdot \sqrt{14}}{\sqrt{6}} = \sqrt{\frac{21 \cdot 14}{6}} = \sqrt{\frac{294}{6}} = \sqrt{49} = 7\) 4. \((\sqrt{27} - \sqrt{3}) \cdot \sqrt{3} = \sqrt{27 \cdot 3} - \sqrt{3 \cdot 3} = \sqrt{81} - \sqrt{9} = 9 - 3 = 6\) 5. \((\sqrt{17} - 3) \cdot (\sqrt{17} + 3) = (\sqrt{17})^2 - 3^2 = 17 - 9 = 8\) 6. \(\frac{5 \sqrt{7} \cdot 2 \sqrt{2} \cdot \sqrt{22}}{?} = 10 \sqrt{7 \cdot 2 \cdot 22} = 10 \sqrt{308}\) 7. \(2\sqrt{13} \cdot 5\sqrt{2} \cdot \sqrt{26} = 10 \sqrt{13 \cdot 2 \cdot 26} = 10 \sqrt{676} = 10 \cdot 26 = 260\) 8. \((\sqrt{12} + \sqrt{3}) \cdot \sqrt{3} = \sqrt{12 \cdot 3} + \sqrt{3 \cdot 3} = \sqrt{36} + \sqrt{9} = 6 + 3 = 9\) 9. \((\sqrt{23} - 2) \cdot (\sqrt{23} + 2) = (\sqrt{23})^2 - 2^2 = 23 - 4 = 19\) 10. \(\frac{\sqrt{32} \cdot \sqrt{6}}{\sqrt{12}} = \sqrt{\frac{32 \cdot 6}{12}} = \sqrt{\frac{192}{12}} = \sqrt{16} = 4\) 11. \(7\sqrt{15} \cdot 2\sqrt{2} \cdot \sqrt{30} = 14\sqrt{15 \cdot 2 \cdot 30} = 14\sqrt{900} = 14 \cdot 30 = 420\) 12. \(\frac{\sqrt{35} \cdot \sqrt{21}}{\sqrt{15}} = \sqrt{\frac{35 \cdot 21}{15}} = \sqrt{\frac{735}{15}} = \sqrt{49} = 7\) (Letter A) 13. \(4\sqrt{17} \cdot 5\sqrt{2} \cdot \sqrt{34} = 20\sqrt{17 \cdot 2 \cdot 34} = 20\sqrt{1156} = 20 \cdot 34 = 680\) 14. \(\frac{\sqrt{20} \cdot \sqrt{32}}{\sqrt{10}} = \sqrt{\frac{20 \cdot 32}{10}} = \sqrt{\frac{640}{10}} = \sqrt{64} = 8\) (Letter C) 15. \(5\sqrt{7} \cdot 6\sqrt{3} \cdot \sqrt{21} = 30\sqrt{7 \cdot 3 \cdot 21} = 30\sqrt{441} = 30 \cdot 21 = 630\) 16. \((\sqrt{18} - \sqrt{2}) \cdot \sqrt{2} = \sqrt{18 \cdot 2} - \sqrt{2 \cdot 2} = \sqrt{36} - \sqrt{4} = 6 - 2 = 4\) 17. \((\sqrt{11} - 3) \cdot (\sqrt{11} + 3) = (\sqrt{11})^2 - 3^2 = 11 - 9 = 2\)

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Ответ:

Решение:

  1. \(\sqrt{\frac{3 \cdot 42}{70}} = \sqrt{\frac{126}{70}} = \sqrt{1.8}\) (Запись \(\sqrt{0}=3\) и \(\frac{3 · 42}{70}\) некорректна. Правильный результат \(\sqrt{1.8}\), или \(\frac{3\sqrt{70}}{10}\) )
  2. \(\frac{\sqrt{15} \cdot \sqrt{12}}{\sqrt{20}} = \sqrt{\frac{15 \cdot 12}{20}} = \sqrt{\frac{180}{20}} = \sqrt{9} = 3\)
  3. \(\frac{\sqrt{21} \cdot \sqrt{14}}{\sqrt{6}} = \sqrt{\frac{21 \cdot 14}{6}} = \sqrt{\frac{294}{6}} = \sqrt{49} = 7\)
  4. \((\sqrt{27} - \sqrt{3}) \cdot \sqrt{3} = \sqrt{27 \cdot 3} - \sqrt{3 \cdot 3} = \sqrt{81} - \sqrt{9} = 9 - 3 = 6\)
  5. \((\sqrt{17} - 3) \cdot (\sqrt{17} + 3) = (\sqrt{17})^2 - 3^2 = 17 - 9 = 8\)
  6. \(\frac{5 \sqrt{7} \cdot 2 \sqrt{2} \cdot \sqrt{22}}{?} \) — условие неполное. Если предполагается \(5\sqrt{7} \cdot 2\sqrt{2} \cdot \sqrt{22}\), то: \(10 \sqrt{7 \cdot 2 \cdot 22} = 10 \sqrt{308} = 10 \sqrt{4 \cdot 77} = 10 \cdot 2 \sqrt{77} = 20 \sqrt{77}\).
  7. \(2\sqrt{13} \cdot 5\sqrt{2} \cdot \sqrt{26} = 10 \sqrt{13 \cdot 2 \cdot 26} = 10 \sqrt{26 \cdot 26} = 10 \cdot 26 = 260\)
  8. \((\sqrt{12} + \sqrt{3}) \cdot \sqrt{3} = \sqrt{12 \cdot 3} + \sqrt{3 \cdot 3} = \sqrt{36} + \sqrt{9} = 6 + 3 = 9\)
  9. \((\sqrt{23} - 2) \cdot (\sqrt{23} + 2) = (\sqrt{23})^2 - 2^2 = 23 - 4 = 19\)
  10. \(\frac{\sqrt{32} \cdot \sqrt{6}}{\sqrt{12}} = \sqrt{\frac{32 \cdot 6}{12}} = \sqrt{\frac{192}{12}} = \sqrt{16} = 4\)
  11. \(7\sqrt{15} \cdot 2\sqrt{2} \cdot \sqrt{30} = 14\sqrt{15 \cdot 2 \cdot 30} = 14\sqrt{900} = 14 \cdot 30 = 420\)
  12. \(\frac{\sqrt{35} \cdot \sqrt{21}}{\sqrt{15}} = \sqrt{\frac{35 \cdot 21}{15}} = \sqrt{\frac{735}{15}} = \sqrt{49} = 7\) (Отметка А)
  13. \(4\sqrt{17} \cdot 5\sqrt{2} \cdot \sqrt{34} = 20\sqrt{17 \cdot 2 \cdot 34} = 20\sqrt{1156} = 20 \cdot 34 = 680\)
  14. \(\frac{\sqrt{20} \cdot \sqrt{32}}{\sqrt{10}} = \sqrt{\frac{20 \cdot 32}{10}} = \sqrt{\frac{640}{10}} = \sqrt{64} = 8\) (Отметка С)
  15. \(5\sqrt{7} \cdot 6\sqrt{3} \cdot \sqrt{21} = 30\sqrt{7 \cdot 3 \cdot 21} = 30\sqrt{441} = 30 \cdot 21 = 630\)
  16. \((\sqrt{18} - \sqrt{2}) \cdot \sqrt{2} = \sqrt{18 \cdot 2} - \sqrt{2 \cdot 2} = \sqrt{36} - \sqrt{4} = 6 - 2 = 4\)
  17. \((\sqrt{11} - 3) \cdot (\sqrt{11} + 3) = (\sqrt{11})^2 - 3^2 = 11 - 9 = 2\)

Ответ: 1. \(\sqrt{1.8}\), 2. 3, 3. 7, 4. 6, 5. 8, 6. \(20\sqrt{77}\) (при условии, что \(? = 1\)), 7. 260, 8. 9, 9. 19, 10. 4, 11. 420, 12. 7 (А), 13. 680, 14. 8 (С), 15. 630, 16. 4, 17. 2.

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