\(A=\frac{\sqrt{\left(x-1\right)+2\sqrt{x-1}+1}+\sqrt{\left(x-1\right)-2\sqrt{x-1}+1}}{\sqrt{\left(\sqrt{\left(x-1\right)+2\sqrt{x-1}+1}\right)}-\sqrt{\left(x-1\right)-2\sqrt{x-1}+1}}\)=\(\frac{\sqrt{\left(\sqrt{x-1}+1\right)^2}+\sqrt{\left(\sqrt{x-1}-1\right)^2}}{\sqrt{\left(\sqrt{x-1}+1\right)^2}-\sqrt{\left(\sqrt{x-1}-1\right)^2}}\)Vì x>/2

=\(\frac{\sqrt{x-1}+1+\sqrt{x-1}-1}{\sqrt{x-1}+1-\sqrt{x-1}+1}=\frac{2\sqrt{x-1}}{2}=\sqrt{x-1}\)

\(\frac{1}{\sqrt{2}}.A=\frac{\sqrt{\left(x-1\right)+2\sqrt{x-1}+1}+\sqrt{\left(x-1\right)-2\sqrt{x-1}+1}}{\sqrt{\left(2x-1\right)+2\sqrt{2x-1}+1}-\sqrt{\left(2x-1\right)-2\sqrt{2x-1}+1}}\)

                   \(=\frac{\sqrt{\left[\left(\sqrt{x-1}+1\right)\right]^2}+\sqrt{\left[\left(\sqrt{x-1}-1\right)^2\right]}}{\sqrt{\left[\sqrt{2x-1}+1\right]^2}-\sqrt{\left[\left(\sqrt{2x-1}\right)-1\right]^2}}\)

                     \(=\frac{\left|\sqrt{x-1}+1\right|+\left|\sqrt{x-1}-1\right|}{\left|\sqrt{2x-1}+1\right|-\left|\sqrt{2x-1}-1\right|}\)

DO X>2 NÊN TOÀN BỘ BIỂU THỨC TRONG TRỊ TUYỆT ĐỐI ĐỀU DƯƠNG 

\(\frac{1}{\sqrt{2}}.A=\frac{2\sqrt{x-1}}{2}=\sqrt{x-1}\)

=>\(A=\frac{\sqrt{x-1}}{\sqrt{2}}\)

ĐKXĐ: x>=0; x<>1

a: \(A=\frac{x\cdot\sqrt{x}+x}{\sqrt{x}+1}-\frac{x-2\sqrt{x}+1}{\sqrt{x}-1}\)

\(=\frac{x\left(\sqrt{x}+1\right)}{\sqrt{x}+1}-\frac{\left(\sqrt{x}-1\right)^2}{\sqrt{x}-1}\)

\(=x-\left(\sqrt{x}-1\right)=x-\sqrt{x}+1\)

b: A=4

=>\(x-\sqrt{x}+1=4\)

=>\(x-\sqrt{x}-3=0\)

=>\(x-\sqrt{x}+\frac14-\frac{13}{4}=0\)

=>\(\left(\sqrt{x}-\frac12\right)^2=\frac{13}{4}\)

=>\(\sqrt{x}-\frac12=\frac{\sqrt{13}}{2}\)

=>\(\sqrt{x}=\frac{\sqrt{13}+1}{2}\)

=>\(x=\frac{\left(\sqrt{13}+1\right)^2}{4}=\frac{4+2\sqrt{13}}{4}=\frac{2+\sqrt{13}}{2}\)

c: \(A=x-\sqrt{x}+1\)

\(=x-\sqrt{x}+\frac14+\frac34=\left(\sqrt{x}-\frac12\right)^2+\frac34\ge\frac34>0\forall x\) thỏa mãn ĐKXĐ

=>x>=0 và x<>1

a) \(\frac{x\sqrt{x}+y\sqrt{y}}{\sqrt{x}+\sqrt{y}}-\left(\sqrt{x}-\sqrt{y}\right)^2\)

\(=\frac{\left(\sqrt{x}+\sqrt{y}\right)\left(x-\sqrt{xy}+y\right)}{\sqrt{x}+\sqrt{y}}-\left(x-2\sqrt{xy}+y\right)\)

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

b) \(\sqrt{\frac{x-2\sqrt{x}+1}{x+2\sqrt{x}+1}}=\sqrt{\frac{\left(\sqrt{x}-1\right)^2}{\left(\sqrt{x}+1\right)^2}}=\frac{\left|\sqrt{x}-1\right|}{\sqrt{x}+1}\)

c) \(4x-\sqrt{8}+\frac{\sqrt{x^3+2x^2}}{\sqrt{x+2}}=4x-\sqrt{8}+\frac{\sqrt{x^2\left(x+2\right)}}{x+2}=4x-\sqrt{8}+x=5x-\sqrt{8}\)