Ćwiczenie 4/40 na dziś pilne.... Question from @sheesh20

$a) \ (\sqrt{10})^{6} =(10^{\frac{1}{2}})^{6} = 10^{\frac{1}{2}\cdot6} = 10^{3} = 1000\\\\b) \ \sqrt{4^{10}} = (4^{10})^{\frac{1}{2}} = 4^{10\cdot\frac{1}{2}}=4^{5} = 1024\\\\c) \ \sqrt[3]{125^{2}} = (125^{2})^{\frac{1}{3}} = 125^{\frac{2}{3}} = (5^{3})^{\frac{2}{3}} = 5^{2} = 25\\\\d) \ \sqrt[5]{32^{3}} = (32^{3})^{\frac{1}{5}} = 32^{\frac{3}{5}}=(2^{5})^{\frac{3}{5}} = 2^{3} = 8$

$e) \ (\sqrt[8]{625})^{2}=(625^{\frac{1}{8}})^{2} = 625^{\frac{1}{4}} = (5^{4})^{\frac{1}{4}} = 5$

$f) \ \sqrt\sqrt{81}}=\sqrt{9} = 3$

$g) \ (\sqrt[3]{\sqrt{27}})^{2} = ((27^{\frac{1}{2}})^{\frac{1}{3}})^{2} = 27^{\frac{1}{3}} = (3^{3})^{\frac{1}{3}} = 3\\\\h) \ \sqrt[3]{\sqrt{4^{3}}} = ((4^{3})^{\frac{1}{2}})^{\frac{1}{3}} = 4^{\frac{1}{2}} = (2^{2})^{\frac{1}{2}} = 2$

$i) \ (\sqrt{\sqrt[3]{49}})^{3} = ((49^{\frac{1}{3}})^{\frac{1}{2}})^{3} = 49^{\frac{1}{2}} = (7^{2})^{\frac{1}{2}} = 7\\\\j) \ \sqrt[3]{\sqrt[4]{16^{6}}}=((16^{6})^{\frac{1}{4}})^{\frac{1}{3}} = 16^{\frac{1}{2}} = (4^{2})^{\frac{1}{2}} = 4$

Wyjaśnienie:

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

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