Đáp án: $2017$
Giải thích các bước giải:
Ta có:
$x=\sqrt[3]{9+4\sqrt5}+\sqrt[3]{9-4\sqrt5}$
$\to x^3=(\sqrt[3]{9+4\sqrt5}+\sqrt[3]{9-4\sqrt5})^3$
$\to x^3=(9+4\sqrt5)+(9-4\sqrt5) +3\sqrt[3]{9+4\sqrt5}\cdot \sqrt[3]{9-4\sqrt5}(\sqrt[3]{9-4\sqrt5}+\sqrt[3]{9-4\sqrt5})$
$\to x^3=18 +3\sqrt[3]{(9+4\sqrt5)(9-4\sqrt5)}x$
$\to x^3=18 +3\sqrt[3]{81-80}x$
$\to x^3=18 +3x$
$\to x^3-3x=18$
Lại có:
$y=\sqrt[3]{3+2\sqrt2}+\sqrt[3]{3-2\sqrt2}$
$\to y^3=6+3y$
$\to y^3-3y=6$
$\to x^3-3x+y^3-3y+1993=18+6+1993$
$\to x^3+y^3-3(x+y)+1993=2017$
