Express each of the following in the form (a + ib):

Question:

Express each of the following in the form (a + ib):

$\frac{(1-i)^{3}}{\left(1-i^{3}\right)}$

 

Solution:

Given: $\frac{(1-i)^{3}}{\left(1-i^{3}\right)}$

The above equation can be re-written as

$=\frac{(1)^{3}-(i)^{3}-3(1)^{2}(i)+3(1)(i)^{2}}{\left(1-i \times i^{2}\right)}$

$\left[\because(a-b)^{3}=a^{3}-b^{3}-3 a^{2} b+3 a b^{2}\right]$

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

$=\frac{1-i \times i^{2}-3 i+3(-1)}{(1+i)}$

$=\frac{1-i(-1)-3 i-3}{1+i}$

$=\frac{-2+i-3 i}{1+i}$

$=\frac{-2-2 i}{1+i}$

$=\frac{-2(1+i)}{1+i}$

 

$=-2+0 i$

 

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