Integral Powers of a Complex Number

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 non-zero 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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