Question

Преобразуйте выражение к виду $sqrt[n]{x} sqrt{2m sqrt[3]{ frac{1}{4 m^{2} } sqrt{ frac{n}{m} } } } : sqrt[12]{nm}$

Answer 1

$sqrt{2m sqrt[3]{frac{1}{4m^2} sqrt{frac{n}{n}}} }: sqrt[12]{nm} = sqrt{2mleft (frac{1}{4m^2}cdot (frac{n}{m})^{frac{1}{2}}right )^{frac{1}{3}}} :(nm)^{frac{1}{12}}=\\=sqrt{frac{2mcdot n^{frac{1}{6}}}{2^{frac{2}{3}}cdot m^frac{2}{3}cdot m^{frac{1}{6}}}}}:(nm)^{frac{1}{12}}=sqrt{ frac{2^{frac{1}{3}}cdot m^frac{1}{3}n^{frac{1}{6}}}{m^{frac{1}{6}}}}:(nm)^{frac{1}{12}}=$
 
 $=left(2^{frac{1}{3}}cdot m^{frac{1}{6}}cdot n^{frac{1}{6}}right )^{frac{1}{2}}:(nm)^{frac{1}{12}}=$ $frac{2^{frac{1}{6}}cdot (nm)^{frac{1}{12}}}{(nm)^{frac{1}{12}}}=2^{frac{1}{6}}=sqrt[6]{2}$