Oblicz ...- Zadanie 80: Matematyka z plusem 1. Zakres podstawowy i rozszerzony

Rozwiązanie

$4^{\frac{1}{2}} = (2^{2})^{\frac{1}{2}} = 2^{2 \cdot \frac{1}{2}} = 2^{1} = 2$

$8^{- \frac{1}{3}} = (2^{3})^{- \frac{1}{3}} = 2^{3 \cdot (- \frac{1}{3})} = 2^{- 1} = \frac{1}{2}$

$27^{\frac{2}{3}} = (3^{3})^{\frac{2}{3}} = 3^{3 \cdot \frac{2}{3}} = 3^{2} = 9$

$32^{1, 6} = (2^{5})^{\frac{16}{10}} = (2^{5})^{\frac{8}{5}} = 2^{5 \cdot \frac{8}{5}} = 2^{8} = 256$

$81^{- 1, 25} = (3^{4})^{- \frac{5}{4}} = 3^{4 \cdot (- \frac{5}{4})} = 3^{- 5} = (\frac{1}{3})^{5} = \frac{1}{3 ^{5}} = \frac{1}{243}$

$9^{2, 5} = (3^{2})^{\frac{5}{2}} = 3^{2 \cdot \frac{5}{2}} = 3^{5} = 243$

$0, 01^{- 1, 5} = (\frac{1}{100})^{- \frac{3}{2}} = (\frac{1}{10 ^{2}})^{- \frac{3}{2}} = ((\frac{1}{10})^{2})^{- \frac{3}{2}} = (\frac{1}{10})^{2 \cdot (- \frac{3}{2})} = (\frac{1}{10})^{- 3} =$

$= 10^{3} = 1000$

$0, 0016^{0, 75} = (\frac{16}{10 000})^{\frac{3}{4}} = (\frac{2 ^{4}}{10 ^{4}})^{\frac{3}{4}} = ((\frac{2}{10})^{4})^{\frac{3}{4}} = (\frac{2}{10})^{4 \cdot \frac{3}{4}} = (\frac{2}{10})^{3} =$

$= (\frac{1}{5})^{3} = \frac{1}{5 ^{3}} = \frac{1}{125}$

$(3 \frac{3}{8})^{\frac{4}{3}} = (\frac{27}{8})^{\frac{4}{3}} = (\frac{3 ^{3}}{2 ^{3}})^{\frac{4}{3}} = ((\frac{3}{2})^{3})^{\frac{4}{3}} = (\frac{3}{2})^{3 \cdot \frac{4}{3}} = (\frac{3}{2})^{4} = \frac{3 ^{4}}{2 ^{4}} = \frac{81}{16} = 5 \frac{1}{16}$

$(5 \frac{1}{16})^{- 1 \frac{1}{4}} = (\frac{81}{16})^{- \frac{5}{4}} = (\frac{3 ^{4}}{2 ^{4}})^{- \frac{5}{4}} = ((\frac{3}{2})^{4})^{- \frac{5}{4}} = (\frac{3}{2})^{4 \cdot (- \frac{5}{4})} = (\frac{3}{2})^{- 5} = (\frac{2}{3})^{5} =$

$= \frac{2 ^{5}}{3 ^{5}} = \frac{32}{243}$

$100 000^{- 0, 4} = (10^{5})^{- \frac{2}{5}} = 10^{5 \cdot (- \frac{2}{5})} = 10^{- 2} = (\frac{1}{10})^{2} = \frac{1}{10 ^{2}} = \frac{1}{100}$

Agnieszka Sermak

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