[tex](\sqrt[3]x - x\sqrt{x})^{11} = (x^{\frac{1}{3}} - x^{\frac{3}{2}})^{11}[/tex]
[tex]1)\,\,T_{4+1} = C_{11}^{4}\cdot x^{\frac{1}{3}(11-4)}\cdot x^{\frac{2}{3}\cdot 4} = C_{11}^4\cdot x^{\frac{7}{3}}\cdot x^{\frac{8}{4}} = C_{11}^4\cdot x^{\frac{13}{3}}[/tex]
[tex]2)\,\,\,T_{k+1} = C_{10}^{k}\cdot x^{\frac{1}{3}(11-k)}\cdot x^{\frac{3}{2}\cdot k}[/tex]
[tex]3)\,\,\, \frac{1}{3}(11-k)+\frac{3}{2}\cdot k = 6 \Rightarrow 22-2k+9k = 36\\ \Rightarrow 7k = 14 \Rightarrow k = 2 \Rightarrow \boxed{T_{3} = C_{10}^2\cdot x^{6}}[/tex]
[tex]4)\,\,\, 11-k = M_{3} \Rightarrow k = 11 - M_{3} \Rightarrow k \in \{11,8,5,2\}\\ 3k = M_{2}\Rightarrow k\in \{0,2,4,6,8,10,12,...\}\\ \\ \Rightarrow k \in \{2,8\} \Rightarrow 2\text{ termeni rationali,}\,\, 9\text{ termeni irationali}[/tex]
