a) \(\frac{\left(\sqrt{x}\right)^3-\left(\sqrt{y}\right)^3}{\left(\sqrt{x}-\sqrt{y}\right)}-\left(\sqrt{x}-\sqrt{y}\right)^2=\frac{\left(\sqrt{x}-\sqrt{y}\right)\left(x+\sqrt{xy}+y\right)}{\sqrt{x}-\sqrt{y}}-x+2\sqrt{xy}-y\)

\(=3\sqrt{xy}\)

b) \(\frac{x-y}{\sqrt{y}-1}.\sqrt{\frac{\left(\sqrt{y}-1\right)^4}{\left(x-1\right)^4}}=\frac{x-y}{\sqrt{y}-1}.\frac{\left(\sqrt{y}-1\right)^2}{\left(x-1\right)^2}=\frac{\left(x-y\right)\left(\sqrt{y}-1\right)}{\left(x-1\right)^2}\)

a) \(=\frac{\left(\sqrt{x}\right)^3-\left(\sqrt{y}\right)^3}{\sqrt{x}-\sqrt{y}}-\left(\sqrt{x}-\sqrt{y}\right)^2=x+\sqrt{xy}+y-x+2\sqrt{xy}-y=3\sqrt{xy}\)

\(C=\dfrac{\sqrt{y^3}-1}{y+\sqrt{y}+1}-\dfrac{y+3\sqrt{y}+2}{\sqrt{y}+1}\) 

\(C=\dfrac{\sqrt{y^3}-1^3}{y+\sqrt{y}+1}-\dfrac{y+\sqrt{y}+2\sqrt{y}+2}{\sqrt{y}+1}\)

\(C=\dfrac{\left(\sqrt{y}+1\right)\left[\left(\sqrt{y}\right)^2+\sqrt{y}\cdot1+1\right]}{y+\sqrt{y}+1}-\dfrac{\left(\sqrt{y}+1\right)\left(\sqrt{y}+2\right)}{\sqrt{y}+1}\)

\(C=\dfrac{\left(\sqrt{y}+1\right)\left(y+\sqrt{y}+1\right)}{y+\sqrt{y}+1}-\left(\sqrt{y}+2\right)\)

\(C=\sqrt{y}+1-\sqrt{y}-2\)

\(C=-3\)

\(C=\dfrac{\left(\sqrt{y}-1\right)\left(y+\sqrt{y}+1\right)}{y+\sqrt{y}+1}-\dfrac{\left(\sqrt{y}+1\right)\left(\sqrt{y}+2\right)}{\sqrt{y}+1}\)

\(=\sqrt{y}-1-\sqrt{y}-2=-3\)

\(=\frac{\sqrt{y}}{\sqrt{y}-2}\times\frac{\left(\sqrt{y}-2\right)\left(\sqrt{y}+2\right)}{\sqrt{4}\cdot\sqrt{y}}+\frac{\sqrt{y}}{\sqrt{y}+2}\times\frac{\left(\sqrt{y}+2\right)\left(\sqrt{y}-2\right)}{\sqrt{4}\cdot\sqrt{y}}\)

\(=\frac{\sqrt{y}+2}{\sqrt{4}}+\frac{\sqrt{y}-2}{\sqrt{4}}=\frac{2\sqrt{y}}{2}=\sqrt{y}\)

b/ đkxd \(y>0;y\ne4\)

tại  \(y=\frac{1}{4}\)( t/m dkxd )  nên \(P=\sqrt{y}=\sqrt{\frac{1}{4}}=\frac{1}{2}\)

Ta có :

 Đặt A=\(\frac{\sqrt{x}-\sqrt{y}}{xy\sqrt{xy}}:\left(\left(\frac{x+y}{xy}\right).\frac{1}{\left(\sqrt{x}+\sqrt{y}\right)^2}+\frac{2.\left(\sqrt{x}+\sqrt{y}\right)}{\sqrt{xy}.\left(\sqrt{x}+\sqrt{y}\right)^3}\right)\)

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

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

=\(\frac{\sqrt{x}-\sqrt{y}}{xy\sqrt{xy}}:\frac{1}{xy}\)

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

=\(\frac{\sqrt{x}-\sqrt{y}}{\sqrt{xy}}\)

=\(\frac{\sqrt{2-\sqrt{3}}-\sqrt{2+\sqrt{3}}}{\sqrt{\left(2-\sqrt{3}\right)\left(2+\sqrt{3}\right)}}\)

=\(\frac{\sqrt{2-\sqrt{3}}-\sqrt{2+\sqrt{3}}}{\sqrt{4-3}}\)

=\(\sqrt{2-\sqrt{3}}-\sqrt{2+\sqrt{3}}\)

=> \(A^2=\left(\sqrt{2-\sqrt{3}}-\sqrt{2+\sqrt{3}}\right)^2\)

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

           =\(4-2\sqrt{4-3}\)

           =\(4-2\)

           =\(2\)

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