Question

$sqrt{ sqrt{2}-1 } * sqrt[4]{3+2 sqrt{2} }$

Answer 1

$sqrt{ sqrt{2}-1 } * sqrt[4]{3+2 sqrt{2} } =sqrt[4]{ (sqrt{2}-1 )^2} * sqrt[4]{3+2 sqrt{2} }= \ =sqrt[4]{2-2 sqrt{2}+1 }* sqrt[4]{3+2 sqrt{2} }=sqrt[4]{3-2 sqrt{2} }* sqrt[4]{3+2 sqrt{2} }= \ =sqrt[4]{(3-2 sqrt{2})(3+2 sqrt{2})}=sqrt[4]{9-4*2}=sqrt[4]{1}$
$epsilon _k=cos frac{2pi k}{4} +i sinfrac{2pi k}{4}, k=0,1,2,3 \ epsilon _0=cos 0 +i sin0=1 \ epsilon _1=cos frac{pi }{2} +i sinfrac{pi }{2}=i \ epsilon _2=cos pi +i sinpi=-1 \ epsilon _3=cos frac{3pi }{2} +i sinfrac{3pi }{2}=-i$