Oblicz: [tex][\frac{125^\frac{2}{3}-(0,2)^{-1}}{(0,5)^{-2}}*(0,2)^{-3}]^\frac{3}{4}[/tex].... Question from @HeezQ

$\left [ \frac{125^{\frac{2}{3}}-(0,2)^{-1}}{(0,5)^{-2}} * (0.2)^ {-3} \right ]^{\frac{3}{4}} =\left [ \frac{ (5^3)^{\frac{2}{3}}-( \frac{2}{10})^{-1}}{( \frac{5}{10})^{-2}} * ( \frac{2}{10})^ {-3} \right ]^{\frac{3}{4}} = \left [ \frac{ 5^{3*\frac{2}{3}}-( \frac{1}{5})^{-1}}{( \frac{1 }{2})^{-2}} * ( \frac{1 }{5})^ {-3} \right ]^{\frac{3}{4}} =\\\\=\left [ \frac{5^2-5^1}{2^2}*5^3 \right ]^{\frac{3}{4}}=\left [ \frac{25-5}{4}*125 \right ]^{\frac{3}{4}}=\left [ \frac{20}{4} * 5^3\right ]^{\frac{3}{4}}=\left [ 5* 5^3 \right ]^{\frac{3}{4}} = \left [ 5^{1+3} \right ]^{\frac{3}{4}}=\\\\=\left [ 5^4 \right ]^{\frac{3}{4}}=5^{4*\frac{3}{4}}=5^3=125$

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basetla

$[\frac{125^{\frac{2}{3}}-(0,2)^{-1}}{(0,5)^{-\frac{1}{2}}}\cdot(0,2)^{-3}]^{\frac{3}{4}}=\\\\=[\frac{(5^{3})^{\frac{2}{3}}-(\frac{2}{10})^{-1}}{(\frac{5}{10})^{-2}}\cdot(\frac{2}{10})^{-3}]^{\frac{3}{4}}=\\\\=[\frac{5^{2}-(\frac{1}{5})^{-1}}{(\frac{1}{2})^{-2}}\cdot(\frac{1}{5})^{-3}]^{\frac{3}{4}}=\\\\=(\frac{25-5}{2^{2}}\cdot5^{3})^{\frac{3}{4}}=\\\\=(\frac{20}{4}\cdot5^{3})^{\frac{3}{4}}=\\\\=(5\cdot5^{3})^{\frac{3}{4}}=\\\\=(5^{4})^{\frac{3}{4}}=5^{3} = 125$

Wyjaśnienie:

$a^{-n} = (\frac{1}{a})^{n}\\\\(a^{m})^{n} = a^{m\cdot n}$

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HeezQ Dziękuję! :)

basetla :)

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