Explicație pas cu pas:
[tex]a = \frac{ \sqrt{3} + \sqrt{ {3}^{2} } + \sqrt{ {3}^{3} } + ... + \sqrt{ {3}^{10} } }{33 + 11 \sqrt{3} } \\ = \frac{ \sqrt{3} + 3 + 3\sqrt{3} + {3}^{2} + ... + {3}^{4} \sqrt{3} + {3}^{5} }{11(3 + \sqrt{3})} \\ = \frac{(3 + {3}^{2} + {3}^{3} + {3}^{4} + {3}^{5}) + (1 + 3 + {3}^{2} + {3}^{3} + {3}^{4}) \sqrt{3} }{11(3 + \sqrt{3})} \\ = \frac{3(1 + {3}^{1} + {3}^{2} + {3}^{3} + {3}^{4}) + (1 + 3 + {3}^{2} + {3}^{3} + {3}^{4}) \sqrt{3} }{11(3 + \sqrt{3})} \\ = \frac{121(3 + \sqrt{3}) }{11(3 + \sqrt{3}) } = 11[/tex]
