Question

positive integral powers of i, explained ?​

Answer 1

Step-by-step explanation:

Any integral power of i is i or, (-1) or 1.

Integral power of i are defined as:

i\(^{0}\) = 1, i\(^{1}\) = i, i\(^{2}\) = -1,

i\(^{3}\) = i\(^{2}\) ∙ i = (-1)i = -i,

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

i\(^{5}\) = i\(^{4}\) ∙ i = 1 ∙ i = i,

i\(^{6}\) = i\(^{4}\) ∙ i\(^{2}\) = 1 ∙ (-1) = -1, and so on.

i\(^{-1}\) = \(\frac{1}{i}\) = \(\frac{1}{i}\) × \(\frac{i}{i}\) = \(\frac{i}{-1}\) = - i

Remember that \(\frac{1}{i}\) = - i

i\(^{-1}\) = \(\frac{1}{i^{2}}\) = \(\frac{1}{-1}\) = -1

i\(^{-3}\) = \(\frac{1}{i^{3}}\) = \(\frac{1}{i^{3}}\) × \(\frac{i}{i}\) = \(\frac{i}{i^{4}}\) = \(\frac{i}{1}\) = i

i\(^{-4}\) = \(\frac{1}{i^{4}}\) = \(\frac{1}{1}\) = 1, and so on.

Note that i\(^{4}\) = 1 and i\(^{-4}\) = 1. It follows that for any integer k,

i\(^{4k}\) = 1, i\(^{4k + 1}\)= i, i\(^{4k + 2}\) = -1, i\(^{4k + 3}\) = - i.

Answer 2

$\huge \mathtt \orange{ \underline \blue{ \underline \red{☆answer☆}}}$

Integral power of a complex number is also a complex number. In other words any integral power of a complex number can be expressed in the form of A + iB, where A and B are real. If z is any complex number, then positive integral powers of z are defined as z1 = a, z2 = z ∙ z, z3 = z2 ∙ z, z4 = z3 ∙ z and so on.