Răspuns:
6 termeni raționali.
Explicație pas cu pas:
[tex](\sqrt 2 + \sqrt[3]{5})^{30}\\ \\ T_{k+1} = C_{30}^{k}\cdot \sqrt{2}^{30-k}\cdot \sqrt[3]{5}^{k} \\ \\ =C_{30}^k\cdot 2^{\frac{30-k}{2}}\cdot 5^{\frac{k}{3}}\\ \\ \\ \Rightarrow 30-k= M_{2}\quad \text{si}\quad k = M_3\\\\\Rightarrow k = 30 - M_{2} \quad \text{si}\quad k = M_3 \\ \\ \Rightarrow k = 30 - 2a,\,\,\, a\in \mathbb{N}\quad \text{si}\quad k = M_3\\ \\ \Rightarrow 30-2a = M_3 \Rightarrow 2a = 30-M_3 \Rightarrow[/tex]
[tex]\Rightarrow a = \dfrac{30-M_3}{2} \Rightarrow a \in\Big\{\dfrac{30}{2},\dfrac{27}{2},\dfrac{24}{2},\dfrac{21}{2},\dfrac{18}{2},...,\dfrac{3}{2},\dfrac{0}{2}\Big\}\\ \\\\ \text{In multimea }a\text{ de la }30,24,18,12,6,0\,\Rightarrow \boxed{6}\text{ numere naturale.}[/tex]
