Odpowiedź:
[tex]5\sqrt{5} =5\cdot 5^{\frac{1}{2} } =5^{1+\frac{1}{2} } =5^{\frac{3}{2} } \\\\7\sqrt{7} =7\cdot 7^{\frac{1}{2} } =7^{1+\frac{1}{2} } =7^{\frac{3}{2} } \\\\2^{\frac{2}{5} } \cdot 8^{\frac{1}{5} } =2^{\frac{2}{5} } \cdot (2^{3} )^{\frac{1}{5} } =2^{\frac{2}{5} } \cdot 2^{\frac{3}{5} } =2^{\frac{2}{5} +\frac{3}{5} }= 2^{1} =2\\\\3^{\frac{5}{7} } \cdot 9^{\frac{1}{7} }=3^{\frac{5}{7} } \cdot (3^{2} )^{\frac{1}{7} }=3^{\frac{5}{7} } \cdot 3^{\frac{2}{7} }=3^{\frac{5}{7}+\frac{2}{7} } =3^{1} =3[/tex]
[tex]9 ^{\frac{3}{10} } \div 27^{-\frac{4}{5} } =9 ^{\frac{3}{10} } \div 27^{-\frac{8}{10} }=(3^{2} ) ^{\frac{3}{10} } \div (3^{3} )^{-\frac{8}{10} }=3^{\frac{6}{10} } \div 3^{-\frac{24}{10} }=3 ^{\frac{6}{10} -(-\frac{24}{10} )} =3 ^{\frac{6}{10} +\frac{24}{10})}=3 ^{\frac{30}{10} }=3^{3} =27\\\\25 ^{-\frac{2}{7} } \div 125^{\frac{1}{7} } =(2^{2} ) ^{-\frac{2}{7} } \div (5^{3} )^{\frac{1}{7} } =5 ^{-\frac{4}{7} } \div 5^{\frac{3}{7} } =5 ^{-\frac{4}{7} -\frac{3}{7} }=5^{-1} =(\frac{1}{5} )^{1} =\frac{1}{5}[/tex]
Korzystam ze wzorów:
[tex]x^{n} \cdot x^{m} =x^{n+m}\\\\x^{n} \div x^{m} =x^{n-m} \\\\(x^{n} )^{m} =x^{n\cdot m} \\\\x^{1} =x\\\\x^{-n} =(\frac{1}{x} )^{n}[/tex]
