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a: \(=\dfrac{\sqrt{ab}\left(\sqrt{a}-\sqrt{b}\right)}{\sqrt{a}-\sqrt{b}}-\sqrt{ab}=\sqrt{ab}-\sqrt{ab}=0\)
b: \(=\dfrac{\left(\sqrt{x}-2\sqrt{y}\right)^2}{\sqrt{x}-2\sqrt{y}}+\dfrac{\sqrt{y}\left(\sqrt{x}+\sqrt{y}\right)}{\sqrt{x}+\sqrt{y}}\)
\(=\sqrt{x}-2\sqrt{y}+\sqrt{y}=\sqrt{x}-\sqrt{y}\)
c: \(=\sqrt{x}+2-\dfrac{x-4}{\sqrt{x}-2}\)
\(=\sqrt{x}+2-\sqrt{x}-2=0\)
a: \(=x-\sqrt{xy}+y-x+2\sqrt{xy}-y=\sqrt{xy}\)
b: \(=\dfrac{1+\sqrt{a}}{a-\sqrt{a}}\cdot\dfrac{\left(\sqrt{a}-1\right)^2}{\sqrt{a}+1}=\dfrac{\sqrt{a}-1}{\sqrt{a}}\)
a) Rút gọn được \(\dfrac{\sqrt{xy}}{x-\sqrt{xy}+y}\)
c) \(H=\dfrac{\sqrt{xy}}{x-\sqrt{xy}+y}\Rightarrow H^2=\dfrac{xy}{\left(x-\sqrt{xy}+y\right)^2}\)
\(\Rightarrow H^2-H=\dfrac{xy}{\left(x-\sqrt{xy}+y\right)^2}-\dfrac{\sqrt{xy}}{x-\sqrt{xy}+y}=\dfrac{xy-\sqrt{xy}\left(x-\sqrt{xy}+y\right)}{\left(x-\sqrt{xy}+y\right)^2}\)
\(=\dfrac{2xy-x\sqrt{xy}-y\sqrt{xy}}{\left(x-\sqrt{xy}+y\right)^2}=\dfrac{-\sqrt{xy}\left(x-2\sqrt{xy}+y\right)}{\left(x-\sqrt{xy}+y\right)^2}=-\dfrac{\sqrt{xy}\left(\sqrt{x}-\sqrt{y}\right)^2}{\left(x-\sqrt{xy}+y\right)^2}\)
Do \(\left\{{}\begin{matrix}\sqrt{xy}\ge0\\\left(\sqrt{x}-\sqrt{y}\right)^2\ge0\\\left(x-\sqrt{xy}+y\right)^2\ge0\end{matrix}\right.\)
\(\Rightarrow H^2-H=-\dfrac{\sqrt{xy}\left(\sqrt{x}-\sqrt{y}\right)^2}{\left(x-\sqrt{xy}+y\right)^2}\le0\Rightarrow H^2\le H\)
Mà \(H\ge0\left(cmt\right)\Rightarrow H\le\sqrt{H}\)
\(P=\left(\frac{\sqrt{x}+1}{\sqrt{xy}+1}+\frac{\sqrt{xy}+\sqrt{x}}{1-\sqrt{xy}}+1\right):\left(1-\frac{\sqrt{xy}+\sqrt{x}}{\sqrt{xy}-1}-\frac{\sqrt{x}+1}{\sqrt{xy}+1}\right)\)
+) Đặt \(Q=\frac{\sqrt{x}+1}{\sqrt{xy}+1}+\frac{\sqrt{xy}+\sqrt{x}}{1-\sqrt{xy}}+1\)
\(Q=\frac{\left(\sqrt{x}+1\right)\left(\sqrt{xy}-1\right)}{xy-1}-\frac{\left(\sqrt{xy}+\sqrt{x}\right)\left(\sqrt{xy}+1\right)}{xy-1}+\frac{xy-1}{xy-1}\)
\(Q=\frac{x\sqrt{y}-\sqrt{x}+\sqrt{xy}-1-xy-x\sqrt{y}-\sqrt{xy}-\sqrt{x}+xy-1}{xy-1}\)
\(Q=\frac{-2-2\sqrt{x}}{xy-1}\)
\(Q=\frac{-2\left(\sqrt{x}+1\right)}{xy-1}\)
+) Đặt \(K=1-\frac{\sqrt{xy}+\sqrt{x}}{\sqrt{xy}-1}-\frac{\sqrt{x}+1}{\sqrt{xy}+1}\)
\(K=\frac{xy-1}{xy-1}-\frac{\left(\sqrt{xy}+\sqrt{x}\right)\left(\sqrt{xy}+1\right)}{xy-1}-\frac{\left(\sqrt{x}+1\right)\left(\sqrt{xy}-1\right)}{xy-1}\)
\(K=\frac{xy-1-xy-x\sqrt{y}-\sqrt{xy}-\sqrt{x}-x\sqrt{y}+\sqrt{x}-\sqrt{xy}+1}{xy-1}\)
\(K=\frac{-2x\sqrt{y}-2\sqrt{xy}}{xy-1}\)
\(K=\frac{-2\sqrt{xy}\left(\sqrt{x}+1\right)}{xy-1}\)
Ta có : \(P=Q:K\)
\(\Leftrightarrow P=\frac{-2\left(\sqrt{x}+1\right)}{xy-1}:\frac{-2\sqrt{xy}\left(\sqrt{x}+1\right)}{xy-1}\)
\(\Leftrightarrow P=\frac{-2\left(\sqrt{x}+1\right)\left(xy-1\right)}{-2\sqrt{xy}\left(\sqrt{x}+1\right)\left(xy-1\right)}\)
\(\Leftrightarrow P=\frac{1}{\sqrt{xy}}\)
Vậy...
Q= [\(\frac{\left(\sqrt{x}-\sqrt{y}\right)\left(\sqrt{x}+\sqrt{y}\right)}{\sqrt{x}-\sqrt{y}}+\frac{\left(\sqrt{x}-\sqrt{y}\right)\left(x+\sqrt{xy}+y\right)}{\left(\sqrt{y}-\sqrt{x}\right)\left(\sqrt{y}+\sqrt{x}\right)}\)]\(:\frac{x-2\sqrt{xy}+y+\sqrt{xy}}{\sqrt{x}+\sqrt{y}}\)
\(Q=\left(\sqrt{x}+\sqrt{y}-\frac{x+\sqrt{xy}+y}{\sqrt{x}+\sqrt{y}}\right):\frac{x-\sqrt{xy}+y}{\sqrt{x}+\sqrt{y}}\)
\(Q=\frac{x+2\sqrt{xy}+y-x-\sqrt{xy}-y}{\sqrt{x}+\sqrt{y}}.\frac{\sqrt{x}+\sqrt{y}}{x-\sqrt{xy}+y}\)
\(Q=\frac{\sqrt{xy}}{x-\sqrt{xy}+y}\)
Áp dụng Cô-si:
\(x+y\ge2\sqrt{xy}\)
Do đó:
\(H\le\dfrac{\sqrt{xy}}{2\sqrt{xy}-\sqrt{xy}}=1\)
Mà \(x>y\) nên dấu "=" không xảy ra
\(\Rightarrow H< 1\)
Kết hợp dữ kiện đề bài, ta được:
\(\Rightarrow0< H< 1\)
\(\Rightarrow\sqrt{H}< 1\)
Xét:
\(H-\sqrt{H}=\sqrt{H}\left(\sqrt{H}-1\right)< 0\)
\(\Rightarrow H< \sqrt{H}\)
Ta có
\(x+y\ge2\sqrt{xy}\\ \Leftrightarrow x+y\ge\sqrt{xy}+\sqrt{xy}\\ \Leftrightarrow x+y-\sqrt{xy}\ge\sqrt{xy}\\ \Rightarrow\dfrac{\sqrt{xy}}{yx-\sqrt{xy}+y}\)
Có mẫu luôn lớn hơn hoặc = tử số
Bằng khi x = y = 1
\(\Rightarrow H\le\sqrt{H};bằng.khi.x=y=1\)