Express the given complex number in the form $a + i b :$ $i^{- 39}$

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Solution

$i^{- 39} = i^{4 \times (- 9) - 3} = {(i^{4})}^{- 9} . i^{- 3}$
$= {(1)}^{- 9} . i^{- 3}, [∵ i^{4} = 1]$
$= \frac{1}{i^{3}} = \frac{1}{- i}, [∵ i^{3} = - i]$
$= \frac{- 1}{i} \times \frac{i}{i}$
$= \frac{- i}{i^{2}} = \frac{- i}{- 1} = i, [∵ i^{2} = - 1]$
$= a + i b$ , where $a = 0, b = 1$

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