Question

cho A= \(\frac{x+y-2\sqrt{xy}}{x-y}\)

a) CMR A=\(\frac{\sqrt{x}-\sqrt{y}}{\sqrt{x}+\sqrt{y}}\) b) tính a khi x= 3+\(2\sqrt{2}\); y=3-2\(\sqrt{2}\)

Answer 1

a) \(A=\frac{x+y-2\sqrt{xy}}{x-y}\left(ĐK:xy\ge0;x\ne y\right)\)

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

=>đpcm

b) Có: \(x=3+2\sqrt{2}=\left(\sqrt{2}+1\right)^2\)

=>\(\sqrt{x}=\sqrt{2}+1\)

\(y=3-2\sqrt{2}=\left(\sqrt{2}-1\right)^2\)

=>\(\sqrt{y}=\left|\sqrt{2}-1\right|=\sqrt{2}-1\)

Nên: \(A=\frac{\sqrt{2}+1-\sqrt{2}+1}{\sqrt{2}+1+\sqrt{2}-1}=\frac{2}{2\sqrt{2}}=\frac{1}{\sqrt{2}}\)