What is 1/((1-i)^5) ?

The solution to 1/((1-i)^5) is -1/8-1/8 i

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simplify 1/((1-i)^5)

simplify

\frac{1}{( 1 - i ) ^{5}}

- \frac{1}{8} - \frac{1}{8} i

Solution steps

\frac{1}{( 1 - i ) ^{5}}

Expand

(1 - i)^{5} : 4 i - 4

= \frac{1}{4 i - 4}

Rewrite

4 i - 4

in standard complex form:

- 4 + 4 i

= \frac{1}{- 4 + 4 i}

Multiply by the conjugate

\frac{- 4 - 4 i}{- 4 - 4 i}

= \frac{1 \cdot ( - 4 - 4 i )}{( - 4 + 4 i ) ( - 4 - 4 i )}

Simplify

\frac{1 \cdot ( - 4 - 4 i )}{( - 4 + 4 i ) ( - 4 - 4 i )} : \frac{- 4 - 4 i}{32}

= \frac{- 4 - 4 i}{32}

Apply the fraction rule:

\frac{a + bi}{c} = \frac{a}{c} + \frac{b}{c} i

= \frac{- 4}{32} + \frac{- 4}{32} i

Simplify

\frac{- 4}{32} + \frac{- 4}{32} i : - \frac{1}{8} - \frac{1}{8} i

= - \frac{1}{8} - \frac{1}{8} i

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