ĐKXĐ: \(x\ge2\)

\(A=\sqrt{x+2\sqrt{2x-4}}+\sqrt{x-2\sqrt{2x-4}}\)

\(=\sqrt{x-2+2.\sqrt{x-2}.\sqrt{2}+2}+\sqrt{x-2-2.\sqrt{x-2}.\sqrt{2}+2}\)

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

\(=\left|\sqrt{x-2}+\sqrt{2}\right|+\left|\sqrt{x-2}-\sqrt{2}\right|=\sqrt{x-2}+\sqrt{2}+\left|\sqrt{x-2}-\sqrt{2}\right|\)

Xét \(x\ge4\Rightarrow\sqrt{x-2}\ge\sqrt{2}\)

\(\Rightarrow A=\sqrt{x-2}+\sqrt{2}+\sqrt{x-2}-\sqrt{2}=2\sqrt{x-2}\)

Xét \(0\le x< 4\Rightarrow\sqrt{x-2}< \sqrt{2}\)

\(\Rightarrow A=\sqrt{x-2}+\sqrt{2}-\sqrt{x-2}+\sqrt{2}=2\sqrt{2}\)

ĐKXĐ: 

\(2x-4\ge0\)và \(x+2\sqrt{2x-4}\ge0\)và \(x-2\sqrt{2x-4}\ge0\)

<=>\(2x\ge4\)và \(x\ge2\sqrt{2x-4}\)

<=>\(x\ge2\text{ và }x^2\ge8x-16\)

<=>\(x\ge2\text{ và }\left(x-4\right)^2\ge0\)

=>\(x\ge2\)

\(A=\sqrt{x+2\sqrt{2x-4}}+\sqrt{x-2\sqrt{2x-4}}\)

\(=\sqrt{x-2+2.\sqrt{2}\sqrt{x-2}+2}+\sqrt{x-2-2.\sqrt{2}\sqrt{x-2}+2}\)

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

\(=\left|\sqrt{x-2}+2\right|+\left|\sqrt{x-2}-2\right|\)

Với \(\sqrt{x-2}-2>0\) thì \(A=\sqrt{x-2}+2+\sqrt{x-2}-2=2\sqrt{x-2}\)

Với \(\sqrt{x-2}-2

Sửa đề: x-4

\(A=\dfrac{x-2\sqrt{x}+x+4\sqrt{x}+4+2x+8}{x-4}=\dfrac{4x+2\sqrt{x}+12}{x-4}\)

\(A=\sqrt{x+2\sqrt{2x-4}}+\sqrt{x-2\sqrt{2x-4}}\)

\(=\sqrt{x-2+2\sqrt{x-2}\sqrt{2}+2}+\sqrt{x-2-2\sqrt{x-2}\sqrt{2}+2}\)

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

\(=\left|\sqrt{x-2}+\sqrt{2}\right|+\left|\sqrt{x-2}-\sqrt{2}\right|\)

\(\text{Với }\sqrt{x-2}\ge\sqrt{2}\text{ thì : }A=\sqrt{x-2}+\sqrt{2}+\sqrt{x-2}-\sqrt{2}=2\sqrt{x-2}\)

\(\text{Với }\sqrt{x-2}\le\sqrt{2}\text{ thì : }A=\sqrt{x-2}+\sqrt{2}+\sqrt{2}-\sqrt{x-2}=2\sqrt{2}\)

\(A=\sqrt{3+\sqrt{5}}+\sqrt{7-3.\sqrt{5}}-\sqrt{2}\)

\(\sqrt{2}.A=\sqrt{5+2\sqrt{5}.1+1}+\sqrt{9-2.3.\sqrt{5}+5}-2\)

\(\sqrt{2}.A=\sqrt{5}+1+3-\sqrt{5}-2=2\)

\(\Rightarrow A=\sqrt{2}\)

ĐKXĐ: \(\hept{\begin{cases}2x-4\ge0\\x+2.\sqrt{2x-4}\ge0\\x-2\sqrt{2x-4}\end{cases}}\Leftrightarrow x\ge2\)

\(\sqrt{x+2.\sqrt{2x-4}}+\sqrt{x-2.\sqrt{2x-4}}\)

\(=\sqrt{x-2+2.\sqrt{x-2}.\sqrt{2}+2}+\sqrt{x-2-2.\sqrt{x-2}.\sqrt{2}+2}\)

\(=\sqrt{x-2}+\sqrt{2}+\left|\sqrt{x-2}-\sqrt{2}\right|\)

Tự phá trị tuyệt đối

a) \(3\sqrt{2x}-4\sqrt{2x}+8-2\sqrt{x}\)

\(=-\left(4\sqrt{2x}-3\sqrt{2x}\right)+8-2\sqrt{x}\)

\(=-\sqrt{2x}-2\sqrt{x}+8\) 

b) \(3\sqrt{2x}-\sqrt{72x}+3\sqrt{18x}+18\)

\(=3\sqrt{2x}-6\sqrt{2x}+3\cdot3\sqrt{2x}+18\)

\(=3\sqrt{2x}-6\sqrt{2x}+9\sqrt{2x}+18\)

\(=\left(3+9-6\right)\sqrt{2x}+18\)

\(=6\sqrt{2x}+18\)

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#Học tốt

\(A=\sqrt{x+2\sqrt{2x-4}}+\sqrt{x-2\sqrt{2x-4}}\)

\(A^2=x+2\sqrt{2x-4}+x-2\sqrt{2x-4}+2\sqrt{\left(x+2\sqrt{2x-4}\right)\left(x-2\sqrt{2x-4}\right)}\)

\(A^2=2x+2\sqrt{x^2-4\left(2x-4\right)}\)

\(A^2=2x+2\sqrt{x^2-8x+16}\)

\(A^2=2x+2\sqrt{\left(x-4\right)^2}\)

\(A^2=2x+2\left|x-4\right|\)

\(A=\sqrt{2x+2\left|x-4\right|}\)

\(\dfrac{2\left(2-\sqrt{x}\right)}{\left(2+\sqrt{x}\right)\left(2-\sqrt{x}\right)}+\dfrac{2\sqrt{x}\left(2+\sqrt{x}\right)}{\left(2+\sqrt{x}\right)\left(2-\sqrt{x}\right)}-\dfrac{2x}{\left(2+\sqrt{x}\right)\left(2-\sqrt{x}\right)}=\dfrac{4-2\sqrt{x}+4\sqrt{x}+2x-2x}{\left(2+\sqrt{x}\right)\left(2-\sqrt{x}\right)}=\dfrac{4-2\sqrt{x}}{\left(2+\sqrt{x}\right)\left(2-\sqrt{x}\right)}=\dfrac{2\left(2-\sqrt{x}\right)}{\left(2+\sqrt{x}\right)\left(2-\sqrt{x}\right)}=\dfrac{2}{2+\sqrt{x}}\)