complex number with multiple roots

Question

compute (1+i)^3 (√3+i)^2

Answer 1

Given complex number is (1 + i)^3 (√3 + i)^2

To find the roots we have to simplify it.

Therefore (1 + i)^3 (√3 + i)^2 = { 1^3 + i^3 + 3(1^2)(i)[1 + i] } { (√3)^2 + i^2 + 2√3i }

[ Since (a + b)^3 = a^3 + b^3 + 3a^2b(a+ b), (a + b)^2 = a^2 + b^2 + 2ab ]

                                             = { 1 - i + 3i(1 + i) } { 3 -1 + 2√3i }    [ Since i^3 = -i, i^2 = -1 ]

                                             = (1 - i + 3i + 3i^2) (2 + 2√3i)         

                                             = (1 +2i - 3) (2 + 2√3i)                       [ Since i^2 = -1 ]

                                             = (2i - 2) (2 + 2√3i)

Therefore the roots are (2i - 2) and (2 + 2√3i)