Question

Помогите решить примеры:
1) $sqrt[4]{24*54}$
2) $frac{ sqrt[3]{1152} * sqrt[3]{12} }{8}$
3) $(5 sqrt[3]{6 sqrt{32}} -2 sqrt[3]{75 sqrt{50} })^5$

Answer 1

$sqrt[4]{24cdot 54} = sqrt[4]{4cdot6cdot 6cdot9} = sqrt[4]{4cdot9cdot 36} =sqrt{2cdot3cdot 6} =sqrt{6cdot 6} =6$

$dfrac{ sqrt[3]{1152} cdot sqrt[3]{12} }{8} = dfrac{ sqrt[3]{1152cdot 12} }{8} = dfrac{ sqrt[3]{8cdot 144cdot 12} }{8} = dfrac{ sqrt[3]{2^3cdot 12^3} }{8} =dfrac{ 2cdot 12 }{8}=3$

$(5 sqrt[3]{6 sqrt{32}} -2 sqrt[3]{75 sqrt{50} })^5 = (5 sqrt[3]{6cdot4 sqrt{2} } -2 sqrt[3]{75 cdot 5 sqrt{2} })^5 = \ =(5 sqrt[3]{24sqrt{2} } -2 sqrt[3]{375 sqrt{2} })^5 =(5 sqrt[3]{8cdot 3sqrt{2} } -2 sqrt[3]{125cdot3 sqrt{2} })^5 = \ =(5 cdot 2sqrt[3]{3sqrt{2} } -2cdot5 sqrt[3]{3 sqrt{2} })^5 = (10sqrt[3]{3sqrt{2} } -10 sqrt[3]{3 sqrt{2} })^5 =0^5=0$