Simplify each of the following and express it in the form (a + ib) :

Solution:

Given: $(1+i)^{3}-(1-i)^{3} \ldots(i)$

We know that

$(a+b)^{3}=a^{3}+3 a^{2} b+3 a b^{2}+b^{3}$

$(a-b)^{3}=a^{3}-3 a^{2} b+3 a b^{2}-b^{3}$

By applying the formulas in eq. (i), we get

$(1)^{3}+3(1)^{2}(i)+3(1)(i)^{2}+(i)^{3}-\left[(1)^{3}-3(1)^{2}(i)+3(1)(i)^{2}-(i)^{3}\right]$

$=1+3 i+3 i^{2}+i^{3}-\left[1-3 i+3 i^{2}-i^{3}\right]$

$=1+3 i+3 i^{2}+i^{3}-1+3 i-3 i^{2}+i^{3}$

$=6 i+2 i^{3}$

$=6 i+2 i\left(i^{2}\right)$

$=6 i+2 i(-1)\left[\because i^{2}=-1\right]$

$=6 i-2 i$

$=4 i$

$=0+4 i$