Power of i :

A number of the form \(a + \iota b, \) where a and b are real numbers, is defined to be a complex number.

Here \(\iota\) means iota, now to understand the topic we will go further.

\(\large\iota\) is equal to \(\large \sqrt{-1}\)

\(\large{\iota}^2=({\sqrt{-1}})^2=1\)

\(\large{\iota}^3=({\sqrt{-1}})^2 . {\iota}=-1.{\iota}={-\iota}\)

\(\large{\iota}^4=({({\iota})^2})^2={\sqrt{-1}}^2=1\)

\(\large{\iota}^5=({({\iota})^2})^2 .\iota ={\sqrt{-1}}^2 \iota\)

\(\large=({\sqrt{-1}})^2 . {\iota}=-1.-1.{\iota}={\iota}\)

Negative powers of i

i.e Negative powers of iota, we have values

\(\large{\iota}^{-1}=({1 \times \iota \over \iota \times \iota})={ \iota \over -1}=-\iota\)

\(\large{\iota}^{-2}=({1 \over \iota^2})={ 1 \over -1}=-1\)

\(\large{\iota}^{-3}={1 \over \iota^3}={1 \over -\iota}={1 \times \iota \over -\iota \times \iota}= \iota \)

In general, for any integer \(k\), power of \(\iota\) will be

\(i^{4k} = 1\),

\(i^{4k+1} = 1\),

\(i^{4k+2} = -1\),

\(i^{4k+3} = -\iota\),