Potęgi

[amberxx]

Bardzo proszę o pomoc... Z góry ogromnie, ogromnie dziękuję!

1. Przedstaw podane wyrażenie w postaci potęgi:
\((2^{\frac{2}{3}} \cdot 0,25^{ \frac{2}{3} } \cdot \sqrt[3]{4})^{-1} : [( \frac{10}{3}^ \frac{2}{5} \cdot 0,81^{\frac{2}{5}} \cdot \sqrt[5]{100} ]^{ -\frac{1}{3}}\)

2. Oblicz
a) \(\frac{ (\frac{2}{3})^{-3} \cdot (3 \sqrt{2})^{-2} }{9^{- \frac{1}{2}} \cdot (2 \sqrt{3})^{-4} }\)
b) \(\frac{0,16^{- \frac{1}{2}} \cdot (5 \sqrt{5})^{-2} \cdot 0,04 }{(27 \sqrt{8})^{\frac{2}{3} } \cdot 5^{-4}}\)
c) \(\frac{ \sqrt{2 \sqrt[3]{2 \sqrt{2} } } \cdot \sqrt{3\sqrt{3 \sqrt[3]{3} } } }{ \sqrt{6 \sqrt{6} } }\)

Wyniki
1. \(9^{ \frac{1}{5}}\)

2.
a) \(81\)
b) \(\frac{1}{36}\)
c) \(\sqrt[12]{3}\)

[denatlu]

2.a
\(\frac{ (\frac{2}{3})^{-3} \cdot (3 \sqrt{2})^{-2} }{9^{- \frac{1}{2}} \cdot (2 \sqrt{3})^{-4} }= \frac{2^{-3} \cdot 3^3 \cdot 3^{-2} \cdot 2^{-1}}{3^{-1} \cdot 2^{-4} \cdot 3^{-2}}=\frac{2^{-4}\cdot 3}{2^{-4} \cdot 3^{-3}}=\frac{3}{3^{-3}}=3^4=81\)

[sebnorth]

1.
\((2^{\frac{2}{3}} \cdot 0,25^{ \frac{2}{3} } \cdot \sqrt[3]{4})^{-1} : [( \frac{10}{3}^ \frac{2}{5} \cdot 0,81^{\frac{2}{5}} \cdot \sqrt[5]{100} ]^{ -\frac{1}{3}} = (\ast)\)


\((2^{\frac{2}{3}} \cdot 0,25^{ \frac{2}{3} } \cdot \sqrt[3]{4})^{-1} = 2^{\frac{2}{3}} \cdot (2^{-2})^{ \frac{2}{3} } \cdot 2^{ \frac{2}{3} } =\)

\(= 2^{\frac{2}{3}} \cdot 2^{ \frac{-4}{3} } \cdot 2^{ \frac{2}{3} } = 2^{ \frac{2}{3} + \frac{-4}{3} + \frac{2}{3} } = 2^0 = 1\)

\(( \frac{10}{3} )^ \frac{2}{5} \cdot ( \frac{3^4}{10^2} )^{\frac{2}{5}} \cdot 10^{ \frac{2}{5} } =\)

\(= \frac{10^{ \frac{2}{5}}}{3^{ \frac{2}{5}}} \cdot \frac{3^{\frac{8}{5}}}{10^{\frac{4}{5}}} \cdot 10^{ \frac{2}{5} } = 3^{\frac{6}{5}}\)

\((\ast) = 1^{-1} : (3^{\frac{6}{5}})^{ -\frac{1}{3}} = (3^{\frac{6}{5}})^{\frac{1}{3}} = 3^{\frac{2}{5}} = 9^{\frac{1}{5}}\)