Найдите значение выражения 1 1 b a при а = √32 7. 4a 5b 4 5 ив √2

Дано выражение:\[\left(\frac{1}{4a} - \frac{1}{5b}\right) : \left(\frac{b}{4} - \frac{a}{5}\right)\]

Подставим b = 1/\(\sqrt{2}\):\[\left(\frac{1}{4a} - \frac{1}{5(\frac{1}{\sqrt{2}})}\right) : \left(\frac{\frac{1}{\sqrt{2}}}{4} - \frac{a}{5}\right) = \left(\frac{1}{4a} - \frac{\sqrt{2}}{5}\right) : \left(\frac{1}{4\sqrt{2}} - \frac{a}{5}\right)\]

Подставим a = \(\sqrt{32}\) = 4\(\sqrt{2}\):\[\left(\frac{1}{4(4\sqrt{2})} - \frac{\sqrt{2}}{5}\right) : \left(\frac{1}{4\sqrt{2}} - \frac{4\sqrt{2}}{5}\right) = \left(\frac{1}{16\sqrt{2}} - \frac{\sqrt{2}}{5}\right) : \left(\frac{1}{4\sqrt{2}} - \frac{4\sqrt{2}}{5}\right)\]

Приведем к общему знаменателю:\[\left(\frac{5 - 16 \cdot 2}{80\sqrt{2}}\right) : \left(\frac{5 - 4 \cdot 4 \cdot 2}{20\sqrt{2}}\right) = \left(\frac{5 - 32}{80\sqrt{2}}\right) : \left(\frac{5 - 32}{20\sqrt{2}}\right) = \frac{-27}{80\sqrt{2}} : \frac{-27}{20\sqrt{2}} = \frac{-27}{80\sqrt{2}} \cdot \frac{20\sqrt{2}}{-27} = \frac{20}{80} = \frac{1}{4}\]

Ответ: 1/4