Ответ: ( 41 ; 40)

Объяснение:

[tex]\left \{ \begin{array}{l} \sqrt[4]{x+y} +\sqrt[4]{x-y} = 4 \\\\ \sqrt{x+y} - \sqrt{x-y} = 8 \end{array}[/tex]

Учтем ОДЗ :

[tex]\displaystyle \left \{ {{ x -y \geq 0 } \atop {x +y} \geq 0 } \right.[/tex]

Введем замену

[tex]a^2 =\sqrt{x+y} ~ ; ~ a =\sqrt[4]{x+ y} \\\\ b^2 =\sqrt{x-y} ~ ; ~ b =\sqrt[4]{x-y}[/tex]

[tex]\left \{ \begin{array}{l} a+ b = 4 \\\\ a^2 -b^2 = 8 \end{array} \Leftrightarrow \left \{ \begin{array}{l} a = 4-b \\\\ a^2 -b^2 = 8 \end{array}[/tex]

[tex](4-b)^2 - b^2 = 8 \\\\ (4-b - b)(4-b+b) = 8 \\\\ (4-2b)\cdot 4 = 8 \\\\ 4 -2b = 2 \\\\ b =1~ ; ~ a = 4 - b = 3[/tex]

Подставим

[tex]a =\sqrt[4]{x+ y} = 3 \\\\ b =\sqrt[4]{x-y} = 1[/tex]

Получим систему

[tex]\left \{ \begin{array}{l} (\sqrt[4]{x+ y})^4 = 3^4 \\\\ (\sqrt[4]{x-y})^4 = 1^4 \end{array} \right. \Leftrightarrow \oplus \left \{ \begin{array}{l} x+ y = 81 \\\\ \underline{x-y = 1 ~~~ } \end{array}\right. \\\\\\ ~ \hspace{12.5em} 2x = 82 \\\\ ~ \hspace{12.5em} x = 41 ~ ; ~ y = 40[/tex]