State whether the following statement is true or false.
$∣ ∣ {(1 + i)}^{i} ∣ ∣$ = $e^{- (8 n + 1) \frac{π}{4}}$

True

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False

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Solution

The correct option is A True
Write $(1 + i)$ as $\sqrt{2} e^{[\frac{i π}{4}]}$

Therefore

$(1 + i)^{i} = (\sqrt{2})^{i} e^{- π / 4} = [e^{- (8 n + 1) \frac{π}{4}}] \times e^{i \frac{l n 2}{2}}$

So,

$| (1 + i)^{i} | = | [e^{- (8 n + 1) \frac{π}{4}}] \times e^{i \frac{l n 2}{2}} | = [e^{- (8 n + 1) \frac{π}{4}}]$

Hence above statement is true.

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