8. Упростите выражение 7ab/(a+7b) * (a/(7b) - a/(a+7b)) и найдите его значение при а = 7√2+7, b=√2-9.

Упростим выражение: $$\frac{7ab}{a+7b} \cdot \left(\frac{a}{7b} - \frac{a}{a+7b}\right) = \frac{7ab}{a+7b} \cdot \frac{a(a+7b) - 7a b}{7b(a+7b)} = \frac{7ab}{a+7b} \cdot \frac{a^2 + 7ab - 7ab}{7b(a+7b)} = \frac{7ab}{a+7b} \cdot \frac{a^2}{7b(a+7b)} = \frac{7a^3b}{7b(a+7b)^2} = \frac{a^3}{(a+7b)^2}$$ Подставим значения $$a = 7\sqrt{2} + 7$$, $$b = \sqrt{2} - 9$$: $$\frac{a^3}{(a+7b)^2} = \frac{(7\sqrt{2} + 7)^3}{(7\sqrt{2} + 7 + 7(\sqrt{2} - 9))^2} = \frac{(7(\sqrt{2} + 1))^3}{(7\sqrt{2} + 7 + 7\sqrt{2} - 63)^2} = \frac{7^3(\sqrt{2} + 1)^3}{(14\sqrt{2} - 56)^2} = \frac{7^3(\sqrt{2} + 1)^3}{(14(\sqrt{2} - 4))^2} = \frac{7^3(\sqrt{2} + 1)^3}{14^2(\sqrt{2} - 4)^2} = \frac{7^3(\sqrt{2} + 1)^3}{2^2 \cdot 7^2(\sqrt{2} - 4)^2} = \frac{7(\sqrt{2} + 1)^3}{4(\sqrt{2} - 4)^2} = \frac{7(\sqrt{2} + 1)^3}{4(2 - 8\sqrt{2} + 16)} = \frac{7(\sqrt{2} + 1)^3}{4(18 - 8\sqrt{2})} = \frac{7(\sqrt{2} + 1)^3}{8(9 - 4\sqrt{2})}$$ $$(\sqrt{2}+1)^3 = (\sqrt{2})^3 + 3(\sqrt{2})^2(1) + 3(\sqrt{2})(1)^2 + 1^3 = 2\sqrt{2} + 3(2) + 3\sqrt{2} + 1 = 5\sqrt{2} + 7$$ $$\frac{7(5\sqrt{2}+7)}{8(9 - 4\sqrt{2})} = \frac{35\sqrt{2}+49}{72 - 32\sqrt{2}}$$ Ответ: (35√2+49)/(72 - 32√2)