Express the given complex number in the form $a + i b :$ $i^{9} + i^{19}$

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Solution

$i^{9} + i^{19} = i^{4 \times 2 + 1} + i^{4 \times 4 + 3}$
$= {(i^{4})}^{2} . i + {(i^{4})}^{4} . i^{3}$
$= 1 \times i + 1 \times (- i), [∵ i^{4} = 1, i^{2} = - 1]$
$= i + (- i) = 0 = a + i b,$ where, $a = 0, b = 0$

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a + i b

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{(1 - i)}^{4}

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i^{- 39}

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