a) \(=\dfrac{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}{\sqrt{x}}:\dfrac{x\sqrt{x}-x+x-\sqrt{x}+\sqrt{x}-x}{\sqrt{x}\left(x+\sqrt{x}\right)}\)
\(=\dfrac{\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right)}{\sqrt{x}}.\dfrac{\sqrt{x}\left(x+\sqrt{x}\right)}{x\sqrt{x}-x}\)
\(=\left(\sqrt{x}-1\right)\left(\sqrt{x}+1\right).\dfrac{x+\sqrt{x}}{x}\)
\(=\left(\sqrt{x}+1\right).\dfrac{x+\sqrt{x}}{\sqrt{x}}=\dfrac{\left(\sqrt{x}+1\right)\left(x+\sqrt{x}\right)}{x}\)
b) Thay \(x=4-2\sqrt{3}\) vào A có:
\(A=\dfrac{\left(4-2\sqrt{3}+1\right)\left(4-2\sqrt{3}+\sqrt{4-2\sqrt{3}}\right)}{4-2\sqrt{3}}\)
\(=\dfrac{\left(5-2\sqrt{3}\right)\left(4-2\sqrt{3}+\sqrt{\left(1-\sqrt{3}\right)^2}\right)}{4-2\sqrt{3}}\)
\(=\dfrac{\left(5-2\sqrt{3}\right)\left(4-2\sqrt{3}+\sqrt{3}-1\right)}{4-2\sqrt{3}}=\dfrac{\left(5-2\sqrt{3}\right)\left(3-\sqrt{3}\right)}{4-2\sqrt{3}}\)
\(=\dfrac{15-5\sqrt{3}-6\sqrt{3}+6}{4-2\sqrt{3}}=\dfrac{21-11\sqrt{3}}{4-2\sqrt{3}}=\dfrac{\left(21-11\sqrt{3}\right)\left(4+2\sqrt{3}\right)}{\left(4-2\sqrt{3}\right)\left(4+2\sqrt{3}\right)}=\dfrac{\left(21-11\sqrt{3}\right)2\left(2+\sqrt{3}\right)}{16-12}\)
\(=\dfrac{\left(21-11\sqrt{3}\right)2\left(2+\sqrt{3}\right)}{4}=\dfrac{42+21\sqrt{3}-22\sqrt{3}-33}{2}=\dfrac{9-\sqrt{3}}{2}\)
