a: \(A=\left(\dfrac{2\sqrt{a}+6}{a-9}+\dfrac{\sqrt{a}}{\sqrt{a}-3}\right)\cdot\dfrac{\sqrt{a}-2}{a-4}\)
\(=\left(\dfrac{2\left(\sqrt{a}+3\right)}{\left(\sqrt{a}+3\right)\left(\sqrt{a}-3\right)}+\dfrac{\sqrt{a}}{\sqrt{a}-3}\right)\cdot\dfrac{1}{\sqrt{a}+2}\)
\(=\dfrac{\sqrt{a}+2}{\left(\sqrt{a}+3\right)\left(\sqrt{a}+2\right)}=\dfrac{1}{\sqrt{a}+3}\)
b: \(B=\dfrac{1}{\sqrt{x}+1}\left(\dfrac{x\sqrt{x}-1}{\sqrt{x}-1}+\sqrt{x}\right)\)
\(=\dfrac{1}{\sqrt{x}+1}\cdot\left(\dfrac{\left(\sqrt{x}-1\right)\left(x+\sqrt{x}+1\right)}{\sqrt{x}-1}+\sqrt{x}\right)\)
\(=\dfrac{1}{\sqrt{x}+1}\left(x+2\sqrt{x}+1\right)=\dfrac{\left(\sqrt{x}+1\right)^2}{\sqrt{x}+1}=\sqrt{x}+1\)
c: \(C=\dfrac{x\sqrt{x}+8}{x-4}-\dfrac{x+4}{\sqrt{x}-2}\)
\(=\dfrac{\left(\sqrt{x}+2\right)\left(x-2\sqrt{x}+4\right)}{\left(\sqrt{x}+2\right)\left(\sqrt{x}-2\right)}-\dfrac{x+4}{\sqrt{x}-2}\)
\(=\dfrac{x-2\sqrt{x}+4-x-4}{\sqrt{x}-2}=-\dfrac{2\sqrt{x}}{\sqrt{x}-2}\)
