Question

Вычислить $12*( sqrt[3]{3 sqrt{3} } - sqrt[5]{49 sqrt{7} } ):( sqrt{3} + sqrt{7} )-6 sqrt{21}$

Answer 1

$12*( sqrt[3]{3 sqrt{3} }- sqrt[5]{49 sqrt{7} } ):( sqrt{3}+ sqrt{7} )-6 sqrt{21} =$

$=12*frac{sqrt[3]{3^1*3^{frac{1}{2} }}- sqrt[5]{7^2*7^{frac{1}{2} } }}{ sqrt{3}+ sqrt{7} }-6 sqrt{21}=12*frac{sqrt[3]{3^{1+ frac{1}{2} }}- sqrt[5]{7^{2+ frac{1}{2} } }}{ sqrt{3}+ sqrt{7} }-6 sqrt{21}=$

$=12*frac{sqrt[3]{3^{frac{3}{2} }}- sqrt[5]{7^{frac{5}{2} } }}{ sqrt{3}+ sqrt{7} }-6 sqrt{21} =12*frac{(3^{frac{3}{2}})^ frac{1}{3} - (7^{frac{5}{2}})^ frac{1}{5} }{ sqrt{3}+ sqrt{7} }-6 sqrt{21}=$

$=12*frac{3^{frac{3}{2}* frac{1}{3}} - 7^{frac{5}{2}*frac{1}{5}}}{ sqrt{3}+ sqrt{7} }-6 sqrt{21} =12*frac{3^{frac{1}{2}} - 7^{frac{1}{2}}}{ sqrt{3}+ sqrt{7} }-6 sqrt{21}=$

$=6*2*frac{ sqrt{3}- sqrt{7} }{ sqrt{3}+ sqrt{7} }-6* sqrt{21} =6*(2*frac{ sqrt{3}- sqrt{7} }{ sqrt{3}+ sqrt{7} }-sqrt{21})=$

$=6*(2*frac{ (sqrt{3}- sqrt{7})*(sqrt{3}- sqrt{7}) }{ (sqrt{3}+ sqrt{7})*(sqrt{3}- sqrt{7}) }-sqrt{21})=$

$=6*(2*frac{ (sqrt{3}- sqrt{7})^2}{3^2-7^2}-sqrt{21}) =6*(2*frac{ (sqrt{3}- sqrt{7})^2}{9-49}-sqrt{21})=$

$=6*(2*frac{ (sqrt{3})^2-2* sqrt{3}* sqrt{7}+( sqrt{7} )^2}{-40}-sqrt{21})=$

$=6*(-frac{2*(3-2* sqrt{3*7}+7)}{2*20}-sqrt{21}) =6*(-frac{7+3-2* sqrt{21}}{20}-sqrt{21})=$

$=6*(-frac{2*5-2* sqrt{21}}{2*10}-sqrt{21}) =6*(-frac{2*(5-sqrt{21})}{2*10}-sqrt{21})=$

$=6*(-frac{5-sqrt{21}}{10}-sqrt{21})=-6(frac{5-sqrt{21}}{10}+frac{10*sqrt{21}}{10})=$

$=-6*frac{5-sqrt{21}+10 sqrt{21} }{10} =-6*frac{5+9sqrt{21}}{10} =-frac{2*3*(5+9sqrt{21})}{2*5}=$

$=-frac{3*(5+9sqrt{21})}{5} =-frac{3*5+3*9sqrt{21}}{5} =-frac{15+27sqrt{21}}{5}$