$\frac{1}{i ^{11}} - \frac{1}{i ^{21}} = 2 i$

Explanation

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$\frac{1}{i ^{11}} - \frac{1}{i ^{21}} \overset{◯}{1} \frac{1}{- i} - \frac{1}{i} \overset{◯}{2} \frac{2 i}{- i ^{2}} \overset{◯}{3} 2 i$
①	$i^{11} = i^{4 \cdot 2 + 3} = (i^{4})^{2} \cdot i^{3} = 1^{2} \cdot (- i) = - i = - i$ $i^{21} = i^{4 \cdot 5 + 1} = (i^{4})^{5} \cdot i^{1} = 1^{5} \cdot i = i$
②	Subtract $\frac{1}{i}$ from $\frac{1}{- i}$ to get $\frac{2 i}{- i ^{2}}$ . To subtract raitonal expressions, both fractions must have the same denominator. We can create a common denominator by multiplying the first fraction by $i$ and the second by $- i$ . $\frac{1}{- i} - \frac{1}{i} = \frac{1 \cdot i}{( - i ) \cdot i} - \frac{1 \cdot ( - i )}{i \cdot ( - i )} = = \frac{i}{- i ^{2}} - \frac{- i}{- i ^{2}} = \frac{2 i}{- i ^{2}}$
③	$- i^{2} = - (- 1) = 1$

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i111​−i211​=2i

$\frac{1}{i ^{11}} - \frac{1}{i ^{21}} = 2 i$