\(\dfrac{3+\sqrt{3}}{\sqrt{3}}-\dfrac{2}{\sqrt{3}+1}+1\)
\(=\dfrac{\left(3+\sqrt{3}\right)\left(\sqrt{3}+1\right)}{\sqrt{3}\left(\sqrt{3}+1\right)}-\dfrac{2\sqrt{3}}{\sqrt{3}\left(\sqrt{3}+1\right)}+\dfrac{\sqrt{3}\left(\sqrt{3}+1\right)}{\sqrt{3}\left(\sqrt{3}+1\right)}\)
\(=\dfrac{\left(3+\sqrt{3}\right)\left(\sqrt{3}+1\right)-2\sqrt{3}+\sqrt{3}\left(\sqrt{3}+1\right)}{\sqrt{3}\left(\sqrt{3}+1\right)}\)
\(=\dfrac{3\sqrt{3}+3+3+\sqrt{3}-2\sqrt{3}+3+\sqrt{3}}{\sqrt{3}\left(\sqrt{3}+1\right)}\)
\(=\dfrac{\left(3\sqrt{3}+\sqrt{3}-2\sqrt{3}+\sqrt{3}\right)+\left(3+3+3\right)}{\sqrt{3}\left(\sqrt{3}+1\right)}\)
\(=\dfrac{3\sqrt{3}+9}{\sqrt{3}\left(\sqrt{3}+1\right)}\)
\(=\dfrac{3\left(\sqrt{3}+3\right)}{3+\sqrt{3}}\)
\(=3\)
\(\dfrac{3+\sqrt{3}}{\sqrt{3}}-\dfrac{2}{\sqrt{3}+1}+1\)
\(=\dfrac{\left(3+\sqrt{3}\right)\left(\sqrt{3}+1\right)-2\sqrt{3}+\sqrt{3}\left(\sqrt{3}+1\right)}{\sqrt{3}\left(\sqrt{3}+1\right)}\)
\(=\dfrac{3\sqrt{3}+3+3+\sqrt{3}-2\sqrt{3}+3+\sqrt{3}}{3+\sqrt{3}}\)
\(=\dfrac{3\sqrt{3}+9}{3+\sqrt{3}}\)
\(=\dfrac{3\left(\sqrt{3}+3\right)}{3+\sqrt{3}}\)
\(=3\)
