Question

Tìm x; y biết \(\dfrac{y^2-x^2}{3}=\dfrac{y^2+x^2}{5}\) và \(x^{10}.y^{10}=1024\)

Answer 1

\(\dfrac{y^2-x^2}{3}=\dfrac{y^2+x^2}{5}=\dfrac{2y^2}{8}=\dfrac{2x^2}{2}\)

\(\Rightarrow\dfrac{2y^2}{8}=\dfrac{2x^2}{2}\Rightarrow y^2=4x^2\)

Lại có \(x^{10}.y^{10}=1024\Leftrightarrow x^{10}.\left(y^2\right)^5=1024\)

\(\Leftrightarrow x^{10}.\left(4x^2\right)^5=1024\Leftrightarrow4^5.x^{10}.x^{10}=1024\)

\(\Leftrightarrow1024.x^{20}=1024\Rightarrow x^{20}=1\Rightarrow x=\pm1\)

\(\Rightarrow y^2=4x^2=4\Rightarrow y=\pm2\)

Vậy \(\left\{{}\begin{matrix}x=\pm1\\y=\pm2\end{matrix}\right.\)