For a positive integer n, find the value of (1-i)^ n (1-1 i )^ n

Question

For a positive integer n, find the value of $(1-\text{i})^{\text{n}}\Big(1-\frac{1}{\text{i}}\Big)^{\text{n}}$

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Answer

We have$(1-\text{i})^{\text{n}}\Big(1-\frac{1}{\text{i}}\Big)^{\text{n}}$
$=\Big[(1-\text{i})^{\text{n}}\Big(1-\frac{1}{\text{i}}\Big)\Big]^{\text{n}}$
$=\Big[(1-\text{i})\Big(1-\frac{1}{\text{i}}\times\frac{\text{i}}{\text{i}}\Big)\Big]^{\text{n}}$
$=\Big[(1-\text{i})\Big(1-\frac{\text{i}}{\text{i}^2}\Big)\Big]^{\text{n}}$ $[\because\text{ i}^2=-1\big]$
$=\big[(1-\text{i})(1+\text{i})\big]^{\text{n}}$
$=\big[1-\text{i}^2\big]^{\text{n}}$
$=\big[1+1\big]^{\text{n}}$
$=2^{\text{n}}$
Hence,$(1-\text{i})^{\text{n}}\Big(1-\frac{1}{\text{i}}\Big)^{\text{n}}=2^{\text{n}}$

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