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\(\(b)\frac{\sqrt{a}+a\sqrt{b}-\sqrt{b}-b\sqrt{a}}{ab-1}\left(a,b\ge0;a,b\ne1\right)\)\)
\(\(=\frac{\left(\sqrt{a}-\sqrt{b}\right)+\left(a\sqrt{b}-b\sqrt{a}\right)}{\left(\sqrt{ab}-1\right)\left(\sqrt{ab+1}\right)}\)\)
\(\(=\frac{\left(\sqrt{a}-\sqrt{b}\right)+\sqrt{ab}\left(\sqrt{a}-\sqrt{b}\right)}{\left(\sqrt{ab}-1\right)\left(\sqrt{ab}+1\right)}\)\)
\(\(=\frac{\left(\sqrt{a}-\sqrt{b}\right)\left(\sqrt{ab}+1\right)}{\left(\sqrt{ab}-1\right)\left(\sqrt{ab}+1\right)}\)\)
\(\(=\frac{\sqrt{a}-\sqrt{b}}{\left(\sqrt{ab}-1\right)}\left(a,b\ge0.a,b\ne1\right)\)\)
_Minh ngụy_
\(\(c)\frac{x\sqrt{x}+y\sqrt{y}}{\sqrt{x}+\sqrt{y}}-\left(\sqrt{x}-\sqrt{y}\right)^2\)\)( tự ghi điều kiện )
\(\(=\frac{x\sqrt{x}+y\sqrt{y}-\left(\sqrt{x}-\sqrt{y}\right)^2.\left(\sqrt{x}+\sqrt{y}\right)}{\sqrt{x}+\sqrt{y}}\)\)
\(\(=\frac{x\sqrt{x}+y\sqrt{y}-\left(x\sqrt{x}+x\sqrt{y}-2x\sqrt{y}-2y\sqrt{x}+y\sqrt{x}+y\sqrt{y}\right)}{\sqrt{x}+\sqrt{y}}\)\)
\(\(=\frac{x\sqrt{y}+y\sqrt{x}}{\sqrt{x}+\sqrt{y}}\)\)( phá ngoặc và tính )
\(\(=\frac{\sqrt{xy}\left(\sqrt{x}+\sqrt{y}\right)}{\sqrt{x}+\sqrt{y}}=\sqrt{xy}\)\)
_Minh ngụy_
2
\(A=\sqrt{1-6x+9x^2}+\sqrt{9x^2-12x+4}\)
A= \(\sqrt{9x^2-6x+1}+\sqrt{9x^2-12x+4}\)
A= \(\sqrt{\left(3x-1\right)^2}+\sqrt{\left(3x-2\right)^2}=\left|3x-1\right|+\left|3x-2\right|\)
ta có |3x-1|+|3x-2|=|3x-1|+|2-3x| ≥ |3x-1+2-3x|=1
=> A ≥ 1
=> Min A =1 khi 1/3 ≤ x ≤ 2/3
\(xy+\sqrt{\left(1+y^2\right)\left(1+x^2\right)}=1\)
\(\Leftrightarrow\sqrt{\left(1+y^2\right)\left(1+x^2\right)}=1-xy\)
\(\Leftrightarrow\left(1+y^2\right)\left(1+x^2\right)=1+x^2y^2-2xy\)
\(\Leftrightarrow1+x^2+y^2+x^2y^2=1+x^2y^2-2xy\)
\(\Leftrightarrow x^2+y^2=-2xy\)
\(\Leftrightarrow x^2+y^2+2xy=0\)
\(\Leftrightarrow\left(x+y\right)^2=0\)
\(\Leftrightarrow x=-y\)
Thay vào ,ta có
\(x\sqrt{1+y^2}+y\sqrt{1+x^2}=-y\sqrt{1+x^2}+y\sqrt{1+x^2}=0\)(đpcm)
đây là cách của mk
@-@
Ta có \(1=\left(xy+\sqrt{\left(1+y^2\right)\left(1+x^2\right)}\right)^2\)
\(=x^2y^2+\left(1+y^2\right)\left(1+x^2\right)+2xy\sqrt{\left(1+y^2\right)\left(1+x^2\right)}\)
\(=x^2y^2+1+x^2+y^2+x^2y^2+2xy\sqrt{\left(1+y^2\right)\left(1+x^2\right)}\)
\(=x^2\left(1+y^2\right)+y^2\left(1+x^2\right)+2xy\sqrt{\left(1+y^2\right)\left(1+x^2\right)}+1\)
\(\Leftrightarrow x^2\left(1+y^2\right)+y^2\left(1+x^2\right)+2xy\sqrt{\left(1+y^2\right)\left(1+x^2\right)}=0\)
\(\Leftrightarrow\left(x\sqrt{1+y^2}+y\sqrt{1+x^2}\right)^2=0\)
\(\Rightarrow x\sqrt{1+y^2}+y\sqrt{1+x^2}=0\)