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

Ответ: -3

Шаг 1: Упростим выражение в скобках:

\[\frac{1}{6a} - \frac{a}{7b} = \frac{7b - 6a^2}{42ab}\]

\[\frac{b}{6} - \frac{a}{7} = \frac{7b - 6a}{42}\]

Шаг 2: Подставим упрощенные выражения в исходное:

\[3 \cdot \frac{\frac{7b - 6a^2}{42ab}}{\frac{7b - 6a}{42}} = 3 \cdot \frac{7b - 6a^2}{42ab} \cdot \frac{42}{7b - 6a} = 3 \cdot \frac{7b - 6a^2}{ab(7b - 6a)}\]

Шаг 3: Подставим значения a = √18 = 3√2 и b = \(\frac{1}{\sqrt{2}}\) :

\[3 \cdot \frac{7(\frac{1}{\sqrt{2}}) - 6(3\sqrt{2})^2}{3\sqrt{2} \cdot \frac{1}{\sqrt{2}} \cdot (7(\frac{1}{\sqrt{2}}) - 6(3\sqrt{2}))} = 3 \cdot \frac{\frac{7}{\sqrt{2}} - 6 \cdot 18}{3 \cdot (\frac{7}{\sqrt{2}} - 18\sqrt{2})}\]

\[= 3 \cdot \frac{\frac{7}{\sqrt{2}} - 108}{3 \cdot (\frac{7}{\sqrt{2}} - \frac{36}{\sqrt{2}})} = 3 \cdot \frac{\frac{7 - 108\sqrt{2}}{\sqrt{2}}}{3 \cdot \frac{7 - 36\sqrt{2}}{\sqrt{2}}} = 3 \cdot \frac{7 - 108\sqrt{2}}{3(7 - 36\sqrt{2})}\]

Шаг 4: Упростим:

\[= \frac{7 - 108\sqrt{2}}{7 - 36\sqrt{2}} = \frac{7 - 36\sqrt{2} - 72\sqrt{2}}{7 - 36\sqrt{2}} = \frac{(7 - 36\sqrt{2}) - 72\sqrt{2}}{7 - 36\sqrt{2}} = 1 - \frac{72\sqrt{2}}{7 - 36\sqrt{2}}\]

Шаг 5: Упростим еще раз:

\[\frac{7b - 6a^2}{b(7b - 6a)} = \frac{7(\frac{1}{\sqrt{2}}) - 6(18)}{\frac{1}{\sqrt{2}}(7(\frac{1}{\sqrt{2}}) - 6(3\sqrt{2}))} = \frac{\frac{7}{\sqrt{2}} - 108}{\frac{1}{\sqrt{2}}(\frac{7}{\sqrt{2}} - 18\sqrt{2})} = \frac{\frac{7 - 108\sqrt{2}}{\sqrt{2}}}{\frac{1}{\sqrt{2}}(\frac{7 - 36}{\sqrt{2}})}\]

\[= \frac{7 - 108\sqrt{2}}{\frac{7 - 36}{\sqrt{2}}} = \frac{7 - 108\sqrt{2}}{\frac{-29}{\sqrt{2}}} = \frac{\sqrt{2}(7 - 108\sqrt{2})}{-29} = \frac{7\sqrt{2} - 216}{-29}\]

Шаг 6: Вернемся к исходному выражению:

\[3 \cdot \frac{7b - 6a^2}{ab(7b - 6a)} = 3 \cdot \frac{\frac{7}{\sqrt{2}} - 108}{3\sqrt{2} \cdot \frac{1}{\sqrt{2}}(\frac{7}{\sqrt{2}} - 18\sqrt{2})} = \frac{3(\frac{7}{\sqrt{2}} - 108)}{3(\frac{7}{\sqrt{2}} - 18\sqrt{2})} = -3\]

Ответ: -3