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Q. $i^{57} + \frac{1}{i^{25}}$, when simplified has the value

BITSATBITSAT 2014

Solution:

$i^{57} + \frac{1}{i^{25}} = \left(i^{4}\right)^{14} . i + \frac{1}{\left(i^{4}\right)^{6} .i} $
$ = i + \frac{1}{i} \left(\because i^{4} = 1\right) $
$ = i - i \left(\because \frac{1}{i} = - i \right) = 0$