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Question
Express the following complex number in the standard form a + i b:
\[\frac{(1 - i )^3}{1 - i^3}\]
Solution
\[ \frac{\left( 1 - i \right)^3}{1 - i^3}\]
\[\frac{\left( 1 + i^2 - 2i \right)\left( 1 - i \right)}{1 - i^3} \left( \because i^2 = - 1 \right)\]
\[\frac{- 2i\left( 1 - i \right)}{1 - i^3}$\times$\frac{1 + i^3}{1 + i^3}\]
\[\frac{- 2i\left( 1 + i^3 - i - i^4 \right)}{1 - i^6}\]
\[\frac{- 2i\left( 1 - i - i - 1 \right)}{1 - i^2}\]
\[\frac{- 2i\left( - 2i \right)}{2}\]
\[ = - 2 + 0i\]
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