Решебник по алгебре и начала математического анализа 10 класс Колягин Задание 532

\[\boxed{\mathbf{532}.}\]

\[\sqrt{65 + 6\sqrt{14}} + \sqrt{65 - 6\sqrt{14}} = A\]

\[x = \sqrt{65 + 6\sqrt{14}};\ \]

\[\ y = \sqrt{65 - 6\sqrt{14}}\]

\[x^{2} + y^{2} = 65 + 6\sqrt{14} + 65 -\]

\[- 6\sqrt{14} = 130;\]

\[xy =\]

\[= \sqrt{\left( 65 + 6\sqrt{14} \right)\left( 65 - 6\sqrt{14} \right)} =\]

\[= \sqrt{4225 - 504} = \sqrt{3721} = 61;\]

\[A^{2} = (x + y)^{2} = x^{2} + 2xy +\]

\[+ y^{2} = 130 + 2 \cdot 61 = 252\]

\[A = \sqrt{252} = 6\sqrt{7}.\]

\[Ответ:6\sqrt{7}.\]