Question

Cho \((X+\sqrt{X^2+2013})(Y+\sqrt{Y+2013})=2013\)

Chứng Minh :\(x^{2013}+y^{2013}=0\)

Answer 1

chỗ kia bạn ghi sai đề r:

mình sửa luôn

\(\left(x+\sqrt{x^2+2013}\right)\left(y+\sqrt{y^2+2013}\right)=2013\)

xét\(\left(x+\sqrt{x^2+2013}\right)\left(y+\sqrt{y^2+2013}\right)\left(\sqrt{y^2+2013}-y\right)=2013\left(\sqrt{y^2+2013}-y\right)\)

\(x+\sqrt{x^2+2013}=\sqrt{y^2+2013}-y\) (1)

xét \(\left(x+\sqrt{x^2+2013}\right)\left(\sqrt{x^2+2013}-x\right)\left(y+\sqrt{y^2+2013}\right)=2013\left(\sqrt{x^2+2013}-x\right)\)

\(y+\sqrt{y^2+2013}=\sqrt{x^2+2013}-x\)(2)

từ (1) và (2)

=> x=-y

nên

\(x^{2013}=-y^{2013}\) hay

\(x^{2013}+y^{2013}=0\)