Calculate sqrt(4+i) . Give your answer in a + bi form. Give the solution in the smallest positive angle.

Question

Calculate sqrt(4+i) . Give your answer in a + bi form. Give the solution in the smallest positive angle.

Answer 1

the answer must be 17^(1/4)e^(.12)i=(17^(1/4))(cos(.12)+isin(.12))

since [17^(1/4)e^(.12)i]²=(√17)e^(.24)i=(√17)(cos.24+isin.24)=4+i

proceeding with

q=(4+i)^(1/2)

same as q=e^(1/2)ln(4+i)

same as q=e^{(1/2)[ln√17+i(arctan(1/4)+2πk)]}

since if z=x+iy then ln(z)=ln(|z|(e^{i(arctan(y/x)+2pik}))=ln|z|+i(arctan(y/x)+2pik)

now q=(17^(1/4))e^i(.12+πk) ,,,set k=0 for smallest positive angle