[tex]x^{5} \cdot 5^{-\frac{2}{5} } =\sqrt{125^{-1} } \cdot 25^{\frac{21}{20} } \\\\x^{5} \cdot 5^{-\frac{2}{5} } =\sqrt{(5^{3} )^{-1} } \cdot (5^{2} )^{\frac{21}{20} } \\\\x^{5} \cdot 5^{-\frac{2}{5} } =\sqrt{5^{3\cdot (-1)} } \cdot 5^{2\cdot \frac{21}{20} } \\\\x^{5} \cdot 5^{-\frac{2}{5} } =\sqrt{5^{-3} } \cdot 5^{ \frac{21}{10} }\\\\x^{5} \cdot 5^{-\frac{2}{5} } =(5^{-3} )^{\frac{1}{2} } \cdot 5^{\frac{21}{10} } \\\\x^{5} \cdot 5^{-\frac{2}{5} } =5^{-\frac{3}{2} } \cdot 5^{\frac{21}{10} }\\\\\\[/tex]
[tex]x^{5} \cdot 5^{-\frac{2}{5} } =5^{-\frac{3}{2} +\frac{21}{10} } \\\\x^{5} \cdot 5^{-\frac{2}{5} } =5^{-\frac{15}{10} +\frac{21}{10} }\\\\x^{5} \cdot 5^{-\frac{2}{5} } =5^{\frac{6}{10} } \\\\x^{5} \cdot 5^{-\frac{2}{5} } =5^{\frac{3}{5} }~~\vert \div 5^{-\frac{2}{5} } \\\\x^{5} =\dfrac{5^{\frac{3}{5} }}{5^{-\frac{2}{5} } } \\\\x^{5} =5^{\frac{3}{5} -(-\frac{2}{5} )} \\\\x^{5} =5^{\frac{3}{5} +\frac{2}{5} }\\\\x^{5} =5^{1}\\\\x^{5} =5~~\Rightarrow~~\sqrt[5]{x^{5} }=\sqrt[5]{5}~~\Rightarrow ~~x=\sqrt[5]{5}[/tex]
Sprawdzam:
[tex]x^{5} \cdot 5^{-\frac{2}{5} } =\sqrt{125^{-1} } \cdot 25^{\frac{21}{20} } \\\\x^{5} \cdot 5^{-\frac{2}{5} } =5^{\frac{3}{5} }\\\\P=5^{\frac{3}{5} }\\\\L=x^{5} \cdot 5^{-\frac{2}{5} } ~~\land ~~x=\sqrt[5]{5} ~~\Rightarrow ~~L=(\sqrt[5]{5})^{5} \cdot 5^{-\frac{2}{5} } \\\\L=(5^{\frac{1}{5} } )^{5} \cdot 5^{-\frac{2}{5} } =5^{1} \cdot 5^{-\frac{2}{5} } =5^{1+(-\frac{2}{5} )} =5^{\frac{3}{5} }\\\\P=5^{\frac{3}{5} }~~\land ~~L= 5^{\frac{3}{5} }~~\Rightarrow ~~L=P\\[/tex]
korzystałam ze wzorów:
[tex]x^{n} \cdot x^{m} =x^{n+m} \\\\x^{n} \div x^{m} =x^{n-m} \\\\\sqrt[n]{x} =x^{\frac{1}{n} } \\\\(x^{n} )^{m} =x^{n\cdot m}[/tex]
