Question

(1 + i) (1 + √3i)^2 divided by (1+i)​

Answer 1

sol:

(1 + i) (1 + √3i)^2 / (1+i)

=(1 + i) ( 1 + 2√3i + 3i²) / (1 + i)

=(1 + i) ( 1 + 2√3i - 3) / (1 + i)............(i²= -1)

=(1 + i) ( -2 + 2√3i) / (1+i)

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

=(-2 + 2√3i -2i + 2√3i²) / (1 + i)

=(-2 + 2√3i -2i - 2√3) / (1 + i)

= -2 - 2√3 + i(2√3 - 2) / (1 + i)

Now, rationalizing,,

= [-2 - 2√3 + i(2√3 - 2)] × (1 - i) / (1 + i) (1 - i)

= 1 [-2 - 2√3 + i(2√3 - 2)] -i[-2 - 2√3 + i(2√3 - 2)] / 1 - i²

= (-2 - 2√3) + i(2√3 - 2) - i(-2 - 2√3) - i²(2√3 - 2)] / 1 + 1

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

= ( -4 + 2√3i - 2i + 2i + 2√3i) / 2

= ( -4 + 4√3i) / 2

= 2 (-2+ 2√3i) / 2

= -2 + 2√3i

Answer is -2 + 2√3i

Amannn's