Zad. 6
[tex]a)\\\\\sqrt[5]9=\sqrt[5]{3^2}=3^{\frac25}\\\\\frac1{\sqrt[3]9}=\frac1{\sqrt[3]{3^2}}=3^{-\frac23}\\\\27\sqrt[3]3=3^3*3^{\frac13}=3^{3+\frac13}=3^{\frac93+\frac1{3}}=3^{\frac{10}3}\\\\9\sqrt[4]{27}=3^2*\sqrt[4]{3^3}=3^2*3^{\frac34}=3^{2+\frac34}=3^{\frac84+\frac34}=3^{\frac{11}4}\\\\\frac{9}{\sqrt[5]3}=3^2:3^{\frac15}=3^{2-\frac15}=3^{\frac{10}5-\frac1{5}}=3^{\frac95}[/tex]
[tex]b)\\\\\sqrt[5]2*\sqrt{\frac12}=2^{\frac15}*2^{-\frac12}=2^{\frac15-\frac12}=2^{\frac{2}{10}-\frac5{10}}=2^{-\frac3{10}}\\\\\frac{\sqrt[3]2}{\sqrt[4]4}=\frac{2^{\frac13}}{(2^2)^{\frac14}}=2^{\frac13-2*\frac14}=2^{\frac13-\frac12}=2^{\frac{2}{6}-\frac36}=2^{-\frac16}\\\\\sqrt8*\sqrt[3]4=2^{3*\frac12}*2^{2*\frac13}=2^{\frac32+\frac23}=2^{\frac{9}{6}+\frac{4}{6}}=2^{\frac{13}6}\\\\[/tex]
[tex]\sqrt[3]{\frac14}*\sqrt[6]2=(2^{-2})^{\frac13}*2^{\frac16}=2^{-\frac23+\frac16}=2^{-\frac{4}{6}+\frac16}=2^{-\frac36}=2^{-\frac12}[/tex]
[tex]\frac{\sqrt[3]2}{\sqrt2}=2^{\frac13}:2^\frac12=2^{\frac13-\frac12}=2^{\frac26-\frac36}=2^{-\frac16}[/tex]
[tex]c)\\\\\sqrt{\sqrt5}=(5^{\frac12})^{\frac12}=5^{\frac14}\\\sqrt[3]{\frac15}=(5^{-1})^{\frac13}=5^{-\frac13}\\\frac1{\sqrt{\sqrt[3]{25}}}=(((5^2)^{\frac13})^{\frac12})^{-1}=5^{2*\frac13*\frac12*(-1)}=5^{-\frac26}=5^{-\frac13}\\(\sqrt[4]{\sqrt[3]5})^2=((5^{\frac13})^{\frac14})^2=5^{\frac2{12}}=5^{\frac16}\\\frac{\sqrt[3]5}{\sqrt{\sqrt5}}=5^{\frac13}:(5^{\frac12})^{\frac12}=5^{\frac13-\frac14}=5^{\frac{4}{12}-\frac{3}{12}}=5^{\frac1{12}}[/tex]
