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a) P = \(\left(\frac{\sqrt{x}}{\sqrt{x}-2}+\frac{\sqrt{x}}{\sqrt{x}+2}\right)\)\(.\)\(\frac{x-4}{\sqrt{4x}}\)
= \(\frac{\sqrt{x}.\left(\sqrt{x}+2\right)+\sqrt{x}.\left(\sqrt{x}-2\right)}{\left(\sqrt{x}-2\right)\left(\sqrt{x}+2\right)}\)\(.\)\(\frac{\left(\sqrt{x}+2\right)\left(\sqrt{x}-2\right)}{\sqrt{4x}}\)
= \(\frac{x+2\sqrt{x}+x-2\sqrt{x}}{\sqrt{4x}}\)
= \(\frac{2x}{2\sqrt{x}}\)= \(\sqrt{x}\)
b) x = \(3-2\sqrt{2}\)=\(2-2\sqrt{2}+1\)= \(\left(\sqrt{2}-1\right)^2\)
Thay x = \(\left(\sqrt{2}-1\right)^2\) vào P ta được
P = \(\sqrt{\left(\sqrt{2}-1\right)^2}\)= \(\sqrt{2}-1\)
a, \(\sqrt{\left(\sqrt{2}\right)^2+2\times2\times\sqrt{2}+2^2}\)+ \(\sqrt{2^2+2\times2\times\sqrt{2}+\left(\sqrt{2}\right)^2}\)
= \(\sqrt{\left(\sqrt{2}+2\right)^2}\)+ \(\sqrt{\left(2-\sqrt{2}\right)^2}\)
= \(\sqrt{2}+2+2-\sqrt{2}\)
= 4
a/ \(=4x-\sqrt{\left(x-2\right)^2}=4x-x+2=3x+2\)
b/ \(=3x+\sqrt{\left(x+3\right)^2}=3x+x+3=4x+3\)
c/ xem lại đb
d/ \(=\frac{\sqrt{\left(x+2\right)^2}}{x+2}=\frac{x+2}{x+2}=1\)
\(A=\frac{x}{y}.\frac{x}{y^2}=\frac{x^2}{y^3}\left(\text{vì }x>0;y< 0\text{ nên: }\frac{x}{y^2}>0\right)\)
\(A=\frac{x}{y}\cdot\sqrt{\frac{x^2}{y^4}}=\frac{x}{y}\cdot\frac{\sqrt{x^2}}{\sqrt{y^4}}=\frac{x}{y}\cdot\frac{\left|x\right|}{\left|y^2\right|}=\frac{x}{y}\cdot\frac{x}{y^2}=\frac{x^2}{y^3}\)( x > 0 ; y < 0 )
a) \(\sqrt{\frac{x-2\sqrt{x}+1}{x+2\sqrt{x}+1}}=\sqrt{\frac{\left(\sqrt{x}-1\right)^2}{\left(\sqrt{x}+1\right)^2}}=\frac{\left|\sqrt{x}-1\right|}{\sqrt{x}+1}\)
b) \(\frac{x-1}{\sqrt{y}-1}\cdot\sqrt{\frac{\left(y-2\sqrt{y}+1\right)^2}{\left(x-1\right)^4}}=\frac{x-1}{\sqrt{y}+1}\cdot\sqrt{\frac{\left(\sqrt{y}-1\right)^4}{\left(x-1\right)^4}}=\frac{x-1}{\sqrt{y}+1}\cdot\frac{\left(\sqrt{y}-1\right)^2}{\left(x-1\right)^2}=\frac{\sqrt{y}-1}{x-1}\)
a)\(\sqrt{\frac{x-2\sqrt{x}+1}{x+2\sqrt{x}+1}}=\sqrt{\frac{\left(\sqrt{x}-1\right)^2}{\left(\sqrt{x}+1\right)^2}}=\frac{\sqrt{\left(\sqrt{x}-1\right)^2}}{\sqrt{\left(\sqrt{x+1}\right)^2}}=\frac{\sqrt{x}-1}{\sqrt{x}+1}\)
b)\(\frac{x-1}{\sqrt{y}-1}\sqrt{\frac{\left(y-2\sqrt{y}+1\right)^2}{\left(x-1\right)^4}}=\frac{x-1}{\sqrt{y}-1}\cdot\frac{\sqrt{\left(\sqrt{y}-1\right)^{2^2}}}{\sqrt{\left(x-1\right)^4}}=\frac{x-1}{\sqrt{y}-1}\cdot\frac{\left(\sqrt{y}-1\right)^2}{\left(x-1\right)^2}=\frac{\sqrt{y}-1}{x-1}\)
a/ \(A=\frac{x}{\left(\sqrt{x}+2\right)\left(\sqrt{x}-2\right)}+\frac{1}{\sqrt{x}-2}+\frac{1}{\sqrt{x}+2}\)
\(=\frac{x+\sqrt{x}+2+\sqrt{x}-2}{\left(\sqrt{x}+2\right)\left(\sqrt{x}-2\right)}=\frac{x+2\sqrt{x}}{\left(\sqrt{x}+2\right)\left(\sqrt{x}-2\right)}\)
\(=\frac{\sqrt{x}\left(\sqrt{x}+2\right)}{\left(\sqrt{x}+2\right)\left(\sqrt{x}-2\right)}=\frac{\sqrt{x}}{\sqrt{x}-2}\)
b/ Thay x = 25 vào A ta được:
\(A=\frac{\sqrt{25}}{\sqrt{25}-2}=\frac{5}{5-2}=\frac{5}{3}\)
c/ A = -1/3 \(\Rightarrow\frac{\sqrt{x}}{\sqrt{x}-2}=-\frac{1}{3}\Rightarrow2-\sqrt{x}=3\sqrt{x}\)
\(\Rightarrow4\sqrt{x}=2\Rightarrow\sqrt{x}=\frac{1}{2}\Rightarrow x=\frac{1}{4}\)
Vậy x = 1/4
\(P=\sqrt{\frac{x^2-4x+4}{2-x}}\left(x\ne2\right)\)
\(=\sqrt{\frac{\left(2-x\right)^2}{2-x}}\)
\(=\sqrt{2-x}\)
Vì \(x^2-4x+4=\left(x-2\right)^2>0\left(\forall x\right)\) nên để căn thức có nghĩa thì
\(\Rightarrow2-x>0\Rightarrow x< 2\)
Ta có:
\(P=\sqrt{\frac{x^2-4x+4}{2-x}}=\sqrt{\frac{\left(2-x\right)^2}{2-x}}=\sqrt{2-x}\)
Vậy \(P=\sqrt{2-x}\)