[tex] x^{\frac{1}{\sqrt{1}+\sqrt{2}}}*x^{\frac{1}{\sqrt{2}+\sqrt{3}}}*x^{\frac{1}{ \sqrt{3}+\sqrt{4}}}*...*x^{\frac{1}{\sqrt{99}+\sqrt{100}}}*x^{\frac{1}{\sqrt{100}+\sqrt{101}}}=[/tex]
amplificam puterile:prima cu [tex]\sqrt{1}-\sqrt{2}[/tex],
a doua cu [tex]\sqrt{2}-\sqrt{3}[/tex]
si tot asa,
iar ulima cu [tex]\sqrt{100}-\sqrt{101}[/tex]
Si obtinem:
[tex]=x^{\frac{\sqrt{1}-\sqrt{2}}{1-2}}}*x^{\frac{\sqrt{2}-\sqrt{3}}{2-3}}*...*x^{\frac{\sqrt{100}-\sqrt{101}}{100-101}}=[/tex]
[tex]\sqrt{2}>\sqrt{1}[/tex]
[tex]\sqrt{3}>\sqrt{2}[/tex]
............................................................
[tex]\sqrt{101}>\sqrt{100}[/tex]
[tex]=x^{\sqrt{2}-\sqrt{1}}*x^{\sqrt{3}-\sqrt{12}}*...*x^{\sqrt{101}-\sqrt{100}}*=[/tex]
[tex]=x^{(\sqrt{2}-\sqrt{1})+(\sqrt{3}-\sqrt{2})+....+(\sqrt{101}-\sqrt{100})}=[/tex]
[tex]=x^{(\sqrt{2}-\sqrt{1}+\sqrt{3}-\sqrt{2}+....+\sqrt{101}-\sqrt{100}}=[/tex]
[tex]=x^{-\sqrt{1}+\sqrt{101}}=[/tex]
([tex]x=(\sqrt[4]{\sqrt[5]{3}})^{\sqrt{101}+1}[/tex]
[tex]x=(\sqrt[20]{3})^{\sqrt{101}+1}[/tex]
[tex]x=({3}^{\frac{1}{20}})^{\sqrt{101}+1}[/tex]
[tex]x=3^{\frac{\sqrt{101}+1}{20}}[/tex] )
[tex]=(3^{\sqrt{101}+1})^{\frac{-\sqrt{1}+\sqrt{101}}{20}}=[/tex]
[tex]=3^{(\sqrt{101}+1)(\frac{-\sqrt{1}+\sqrt{101}}{20})}=[/tex]
[tex]=3^{\frac{(\sqrt{101}+1)(\sqrt{101}-\sqrt{1})}{20}}=[/tex]
[tex]=3^{\frac{101-1}{20}}=[/tex]
[tex]=3^{\frac{100}{20}}=[/tex]
[tex]=3^{5}=[/tex]
[tex]\boxed{=243}[/tex]
Cateva formule o sa-ti prinda bine:
[tex]\boxed{\sqrt[n]{a}=a^{\frac{1}{n}}}[/tex]
[tex]\boxed{\sqrt[m]{\sqrt[n]{a}}=\sqrt[m*n]{a}=a^{\frac{1}{m*n}}}[/tex]
