20. Найдите значение выражения 3 \cdot \frac{1}{6a} : (\frac{b}{6} - \frac{a}{7b}) при а = √18 и b = \frac{1}{\sqrt{2}}.

Ответ: -1/2

1. Упростим выражение: \[3 \cdot \frac{1}{6a} : (\frac{b}{6} - \frac{a}{7b}) = \frac{3}{6a} : (\frac{7b^{2} - 6a}{42b}) = \frac{1}{2a} \cdot \frac{42b}{7b^{2} - 6a} = \frac{21b}{a(7b^{2} - 6a)}\]
2. Подставим значения переменных: \[\frac{21 \cdot \frac{1}{\sqrt{2}}}{\sqrt{18}(7 \cdot \frac{1}{2} - 6\sqrt{18})} = \frac{\frac{21}{\sqrt{2}}}{3\sqrt{2}(\frac{7}{2} - 6 \cdot 3\sqrt{2})} = \frac{\frac{21}{\sqrt{2}}}{3\sqrt{2}(\frac{7}{2} - 18\sqrt{2})} = \frac{\frac{21}{\sqrt{2}}}{3\sqrt{2}(\frac{7 - 36\sqrt{2}}{2})} = \frac{\frac{21}{\sqrt{2}}}{\frac{3\sqrt{2}(7 - 36\sqrt{2})}{2}} = \frac{21}{\sqrt{2}} \cdot \frac{2}{3\sqrt{2}(7 - 36\sqrt{2})} = \frac{42}{6(7 - 36\sqrt{2})} = \frac{7}{7 - 36\sqrt{2}}\]

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