If ( 1-i 1+i )^ 100 =a+ib, find (a, b). - Vidyadip

Question

If $\Big(\frac{1-\text{i}}{1+\text{i}}\Big)^{100}=\text{a}+\text{ib},$ find (a, b).

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Answer

$\Big(\frac{1-\text{i}}{1+\text{i}}\Big)^{100}=\text{a}+\text{ib}$
$\Rightarrow\Big(\frac{(1-\text{i})(1-\text{i})}{(1+\text{i})(1-\text{i})}\Big)^{100}=\text{a}+\text{ib}$ [Rationalizing the denominator]
$\Rightarrow\Big(\frac{(1-2\text{i}-1)}{(1+1)}\Big)^{100}=\text{a}+\text{ib}$
$\Rightarrow\Big(\frac{-2\text{i}}{2}\Big)^{100}=\text{a}+\text{ib}$
$\Rightarrow(-\text{i})^{100}=\text{a}+\text{ib}$
$\Rightarrow1=\text{a}+\text{ib}$
Comparing, we get
$(\text{a},\text{b})=(1,0)$

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