[tex]zad.1\\\\a)\\\\\dfrac{\sqrt{8^{\frac{2}{3} } } }{\sqrt[3]{16} \cdot 4^{\frac{4}{3} } } =\dfrac{(8^{\frac{2}{3} } )^{\frac{1}{2} } }{16^{\frac{1}{3} } \cdot (2^{2} )^{\frac{4}{3} } } =\dfrac{8^{\frac{1}{3} } }{ (2^{4} )^{\frac{1}{3} } \cdot 2^{\frac{8}{3} } } =\dfrac{(2^{3} )^{\frac{1}{3} } }{ 2^{\frac{4}{3} } \cdot 2^{\frac{8}{3} } } =\dfrac{2^{1} }{2^{\frac{4}{3} +\frac{8}{3} } } =\dfrac{2^{1} }{2^{\frac{12}{3} } } =\dfrac{2^{1} }{2^{4} } =2^{1-4} =2^{-3} =(\dfrac{1}{2} )^{3} =\dfrac{1}{8}[/tex]
[tex]Odp:~~Wartosc~~wyrazenia~~wynosi~~2^{-3} .[/tex]
korzystałam ze wzorów:
[tex]x^{n} \cdot x^{m} = x^{n+m} \\\\(x^{n} )^{m} =x^{n\cdot m} \\\\x^{n} \div x^{m} =\dfrac{x^{n} }{x^{m} } =x^{n-m} \\\\x^{0} =1\\\\\sqrt[n]{x} =x^{\frac{1}{n} }[/tex]
[tex]b)~~\sqrt[3]{2\sqrt{2} } =\sqrt[3]{2\cdot 2^{\frac{1}{2} } } =\sqrt[3]{2^{1+\frac{1}{2} } } =\sqrt[3]{2^{\frac{3}{2} } } =(2^{\frac{3}{2} } )^{\frac{1}{3} } =2^{\frac{3}{2} \cdot \frac{1}{3} } =2^{\frac{1}{2} } =\sqrt{2} \\\\Odp:~~Wartosc~~wyrazenia~~wynosi~~2^{\frac{1}{2} } .[/tex]
