[tex] a + \dfrac{1}{a} = -2 \Rightarrow \dfrac{a^{2}}{a} + \dfrac{1}{a} = -2 \Rightarrow a^{2} + 1 = -2a \Rightarrow a^{2}-2a+1=0 \Rightarrow (a-1)^{2} = 0 \Rightarrow \boxed{a=1} \\ \\ a^{2013} + \dfrac{1}{a^{2013}} = 1^{2013} + \dfrac{1}{1^{2013}} = 1 + 1 = 2 \\ \\ \dfrac{-3}{\sqrt{3}} + 2 \sqrt{3} = \dfrac{-3 \sqrt{3}}{3} + 2 \sqrt{3} = \boxed{\sqrt{3}} [/tex]
