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a) Ta có: \(A=\left(\frac{1-x\sqrt{x}}{1-\sqrt{x}}+\sqrt{x}\right)\cdot\left(\frac{1-\sqrt{x}}{1-x}\right)^2\)
\(=\left(\frac{1-x\sqrt{x}+\sqrt{x}\left(1-\sqrt{x}\right)}{1-\sqrt{x}}\right)\cdot\left(\frac{1}{1+\sqrt{x}}\right)^2\)
\(=\frac{1-x\sqrt{x}+\sqrt{x}-x}{1-\sqrt{x}}\cdot\frac{1}{\left(1+\sqrt{x}\right)^2}\)
\(=\frac{-\left(x-1\right)\left(-1-\sqrt{x}\right)}{1-\sqrt{x}}\cdot\frac{1}{\left(1+\sqrt{x}\right)^2}\)
\(=\frac{\left(1+\sqrt{x}\right)\cdot\left(-1-\sqrt{x}\right)}{\left(1+\sqrt{x}\right)^2}\)
\(=\frac{-1\cdot\left(1+\sqrt{x}\right)^2}{\left(1+\sqrt{x}\right)^2}=-1\)
a: \(=x-\sqrt{xy}+y-x+2\sqrt{xy}-y=\sqrt{xy}\)
b: \(=\dfrac{1+\sqrt{a}}{a-\sqrt{a}}\cdot\dfrac{\left(\sqrt{a}-1\right)^2}{\sqrt{a}+1}=\dfrac{\sqrt{a}-1}{\sqrt{a}}\)
Bài 1:
a) \(\left(1+\frac{a+\sqrt{a}}{\sqrt{a}+1}\right)\cdot\left(1-\frac{a-\sqrt{a}}{\sqrt{a}-1}\right)\)
\(=\frac{\sqrt{a}+1+a+\sqrt{a}}{\sqrt{a}+1}\cdot\frac{\sqrt{a}-1-a+\sqrt{a}}{\sqrt{a}-1}\)
\(=\frac{a+2\sqrt{a}+1}{\sqrt{a}+1}\cdot\frac{-a+2\sqrt{a}-1}{\sqrt{a}-1}\)
\(=\frac{\left(\sqrt{a}+1\right)^2}{\sqrt{a}+1}\cdot\frac{-\left(\sqrt{a}-1\right)^2}{\sqrt{a}-1}\)
\(=-\left(\sqrt{a}+1\right)\left(\sqrt{a}-1\right)\)
\(=-\left(a-1\right)\)
\(=1-a\)
b) \(P=\sqrt{x^2+6x+2011}\)
\(P=\sqrt{x^2+6x+9+2002}\)
\(P=\sqrt{\left(x+3\right)^2+2002}\ge\sqrt{2002}\forall x\)
Dấu "=" xảy ra \(\Leftrightarrow x=-3\)
ĐK: \(a\ge0;a\ne1\)
\(\left(1+\frac{a+\sqrt{a}}{\sqrt{a}+1}\right).\left(1+\frac{a-\sqrt{a}}{1-\sqrt{a}}\right)\)
\(=\frac{\sqrt{a}+1+a+\sqrt{a}}{\sqrt{a}+1}.\frac{1-\sqrt{a}+a-\sqrt{a}}{1-\sqrt{a}}\)
\(=\frac{1+2\sqrt{a}+a}{\sqrt{a}+1}.\frac{1-2\sqrt{a}+a}{1-\sqrt{a}}\)
\(=\frac{\left(\sqrt{a}+1\right)^2}{\sqrt{a}+1}.\frac{\left(1-\sqrt{a}\right)^2}{1-\sqrt{a}}\)
\(=\left(\sqrt{a}+1\right)\left(1-\sqrt{a}\right)\)
\(=1-a\)
\(\left(1+\frac{a+\sqrt{a}}{\sqrt{a}+1}\right)\left(1+\frac{a-\sqrt{a}}{1-\sqrt{a}}\right)\)
\(=\left(\frac{\sqrt{a}+1+a+\sqrt{a}}{\sqrt{a}+1}\right)\left(\frac{1-\sqrt{a}+a-\sqrt{a}}{1-\sqrt{a}}\right)\)
\(=\frac{a+2\sqrt{a}+1}{\sqrt{a}+1}.\frac{a-2\sqrt{a}+1}{1-\sqrt{a}}\)
\(=\frac{\left(\sqrt{a}+1\right)^2}{\sqrt{a}+1}.-\frac{\left(\sqrt{a}-1\right)^2}{\sqrt{a}-1}\)
\(=-\left(\sqrt{a}+1\right).\left(\sqrt{a}-1\right)\)
\(=1-a\)
Câu b bạn sửa lại đề
\(a,VT=\left[1+\dfrac{\sqrt{x}\left(\sqrt{x}+1\right)}{\sqrt{x}+1}\right]\left[1-\dfrac{\sqrt{x}\left(\sqrt{x}-1\right)}{\sqrt{x}-1}\right]\\ =\left(1+\sqrt{x}\right)\left(1-\sqrt{x}\right)=1-x=VP\\ b,VT=\dfrac{\sqrt{ab}\left(\sqrt{a}-\sqrt{b}\right)}{\sqrt{ab}}+\dfrac{\left(\sqrt{a}-\sqrt{b}\right)\left(\sqrt{a}+\sqrt{b}\right)}{\sqrt{a}-\sqrt{b}}\\ =\sqrt{a}-\sqrt{b}+\sqrt{a}+\sqrt{b}=2\sqrt{a}=VP\)
a: \(=\left(1+\sqrt{x}\right)\left(1-\sqrt{x}\right)=1-x\)
1)))))))
\(\frac{2}{\sqrt{ab}}:\left(\frac{1}{\sqrt{a}}-\frac{1}{\sqrt{b}}\right)^2-\frac{a+b}{\left(\sqrt{a}-\sqrt{b}\right)^2}\)
\(=\frac{2}{\sqrt{ab}}:\frac{\left(\sqrt{b}-\sqrt{a}\right)^2}{\left(\sqrt{ab}\right)^2}-\frac{a+b}{\left(\sqrt{a}-\sqrt{b}\right)^2}\)
\(=\frac{2}{\sqrt{ab}}.\frac{\left(\sqrt{ab}\right)^2}{\left(\sqrt{a}-\sqrt{b}\right)^2}-\frac{a+b}{\left(\sqrt{a}-\sqrt{b}\right)^2}\)
\(=\frac{2\sqrt{ab}}{\left(\sqrt{a}-\sqrt{b}\right)^2}-\frac{a+b}{\left(\sqrt{a}-\sqrt{b}\right)^2}\)
\(=\frac{2\sqrt{ab}-a-b}{\left(\sqrt{a}-\sqrt{b}\right)^2}\)
\(=\frac{-\left(\sqrt{a}-\sqrt{b}\right)^2}{\left(\sqrt{a}-\sqrt{b}\right)^2}=-1\)
\(\text{VT}=\left(1+\frac{x+\sqrt{x}}{\sqrt{x}+1}\right)\left(1-\frac{x-\sqrt{x}}{\sqrt{x}-1}\right)=\left(1+\frac{\sqrt{x}.\left(\sqrt{x}+1\right)}{\sqrt{x}+1}\right)\left(1-\frac{\sqrt{x}.\left(\sqrt{x}-1\right)}{\sqrt{x}-1}\right)\)
\(=\left(1+\sqrt{x}\right)\left(1-\sqrt{x}\right)=1-x=\text{VP(điều phải chứng minh)}\)
\(A=\left(\frac{a+\sqrt{a}}{\sqrt{a}+1}+1\right).\)\(\left(\frac{a-\sqrt{a}}{\sqrt{a}-1}-1\right)\)
\(=\left(\frac{\sqrt{a}\left(\sqrt{a}+1\right)}{\sqrt{a}+1}+1\right)\)\(\left(\frac{-\sqrt{a}\left(\sqrt{a}-1\right)}{\sqrt{a}-1}-1\right)\)
\(=\left(\sqrt{a}+1\right)\left(-\sqrt{a}-1\right)\)
\(=-\left(\sqrt{a}+1\right)\left(\sqrt{a}+1\right)=-\left(\sqrt{a}+1\right)^2\)
\(b,A=-a^2\Rightarrow-\left(\sqrt{a}+1\right)^2=a^2\)
\(\Leftrightarrow a=\sqrt{a}+1\Rightarrow a-\sqrt{a}-1=0\)
\(\Rightarrow4a-4\sqrt{a}-4=0\)
\(\Rightarrow4a-4\sqrt{a}+1-5=0\)
\(\Rightarrow\left(2\sqrt{a}-1\right)^2-\sqrt{5}^2=0\)
\(\Rightarrow\left(2\sqrt{a}-1+\sqrt{5}\right)\left(2\sqrt{a}-1-\sqrt{5}\right)=0\)
\(\Rightarrow\orbr{\begin{cases}2\sqrt{a}=1-\sqrt{5}\\2\sqrt{a}=1+\sqrt{5}\end{cases}}\)\(\Rightarrow\orbr{\begin{cases}\sqrt{a}=\frac{1-\sqrt{5}}{2}\\\sqrt{a}=\frac{1+\sqrt{5}}{2}\end{cases}}\)
\(\Rightarrow\orbr{\begin{cases}a=\frac{\left(1-\sqrt{5}\right)^2}{4}\left(tm\right)\\a=\frac{\left(1+\sqrt{5}\right)^2}{4}\left(tm\right)\end{cases}}\)
\(\left(\frac{1-a\sqrt{a}}{1-\sqrt{a}}+\sqrt{a}\right)\left(\frac{1-\sqrt{a}}{1-a}\right)^2\)
\(=\frac{1-a\sqrt{a}+\sqrt{a}-a}{1-\sqrt{a}}.\frac{\left(1-\sqrt{a}\right)^2}{\left(1-a\right)^2}\)
\(=\left(1-a\sqrt{a}+\sqrt{a}-a\right)\frac{1-\sqrt{a}}{\left(1-a\right)^2}\)
\(=\frac{1-a\sqrt{a}+\sqrt{a}-a-\sqrt{a}+a.\left(\sqrt{a}\right)^2-\left(\sqrt{a}\right)^2+a\sqrt{a}}{\left(1-a\right)^2}\)
\(=\frac{a^2-2a+1}{\left(1-a\right)^2}=\frac{\left(a-1\right)^2}{\left(1-a\right)^2}\)
\(=\left(\frac{a-1}{1-a}\right)^2=\left(-1\right)^2=1=VP\left(ĐPCM\right)\)