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 z\(^{1}\) = a, z\(^{2}\) = z ∙ z, z\(^{3}\) = z\(^{2}\) ∙ z, z\(^{4}\) = z\(^{3}\) ∙ z and so on.
If z is any nonzero complex number, then negative integral powers of z are defined as:
z\(^{1}\) = \(\frac{1}{z}\), z\(^{2}\) = \(\frac{1}{z^{2}}\), z\(^{3}\) = \(\frac{1}{z^{3}}\), etc.
If z ≠ 0, then z\(^{0}\) = 1.
Integral Power of:
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.
Solved examples on integral powers of a complex number:
1. Express i\(^{109}\) in the form of a + ib.
Solution:
i\(^{109}\)
= i\(^{4 × 27 + 1}\)
= i, [Since, we know that for any integer k, i\(^{4k + 1}\) = i]
= 0 + i, which is the required form of a + ib.
2. Simplify the expression i\(^{35}\) + \(\frac{1}{i^{35}}\) in the form of a + ib.
Solution:
i\(^{35}\) + \(\frac{1}{i^{35}}\)
= i\(^{35}\) + i\(^{35}\)
= i\(^{4 × 8 + 3}\) + i\(^{4 × (9) + 1}\)
= 0 + 0
= 0
= 0 + i0, which is the required form of a + ib.
3. Express (1  i)\(^{4}\) in the standard form a + ib.
Solution:
(1  i)\(^{4}\)
= [(1  i)\(^{2}\)]\(^{2}\)
= [1 + i\(^{2}\)  2i]\(^{2}\)
= (1 + (1) – 2i)\(^{2}\)
= (2i)\(^{2}\)
= 4i\(^{2}\)
= 4(1)
= 4
= 4 + i0, which is the required standard form a + ib.
`11 and 12 Grade Math
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