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

Question

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

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Answer

$(1-\text{i})^\text{n}\big(1-\frac{1}{\text{i}}\big)^\text{n}$
$=(1-\text{i})^\text{n}\big(1-\frac{1}{\text{i}}\big)^\text{n}$
$=\Big\{\frac{(1-\text{i})(\text{i}-1)}{\text{i}}\Big\}^\text{n}$
$=\Big\{\frac{(1-\text{i})(1-\text{i})}{-\text{i}}\Big\}^\text{n}$
$=\Big\{\frac{(1-\text{i})^2}{-\text{i}}\Big\}^\text{n}$
$=\Big\{\frac{(1-2\text{i}-1)}{-\text{i}}\Big\}^\text{n}$
$=\Big\{\frac{-2\text{i}}{-\text{i}}\Big\}^\text{n}=2^\text{n}$

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