Solved Example on Complex Number Power Formula
Example 1: Expand (1 + i)5.
Solution:
Given,
- r = √(12 + 12) = √2
- θ = π/4
Polar form of (1 + i) [Tex]=(2\sqrt{2})^{4}-(\sqrt{2})^{4}i [/Tex]
According to De Moivre’s Theorem
(cosθ + sinθ)n = cos(nθ) + i sin(nθ)
Thus,
(1 + i)5 = [Tex][\sqrt{2}cos(\frac{\pi}{4})+i\ sin(\frac{\pi}{4})]^5 [/Tex]
⇒ (1 + i)5 = [Tex](\sqrt{2})^{5}[cos(\frac{5\pi}{4})+i\ sin(\frac{5\pi}{4})]\\ =(\sqrt{2})^{5}[cos(\pi+\frac{\pi}{4})+i\ sin(\pi+\frac{\pi}{4})]\\ =(\sqrt{2})^{5}[-cos(\frac{\pi}{4})-i\ sin(\frac{\pi}{4})]\\ =(\sqrt{2})^{5}[-\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{2}}i]\\ =(\sqrt{2})^{4}-(\sqrt{2})^{4}i [/Tex]
⇒ (1 + i)5 = −4 − 4i
Example 2: Expand (2 + 2i)6.
Solution:
Here, r = [Tex]\sqrt{(2^2+2^2)} = 2\sqrt{2} [/Tex], θ = π/4
The polar form of (2 + 2i) = [Tex][2\sqrt{2}cos(\frac{\pi}{4})+i\ sin(\frac{\pi}{4})] [/Tex]
According to De Moivre’s Theorem: (cosθ + sinθ)n = cos(nθ) + i sin(nθ).
Thus, (2 + 2i)6 = [Tex][2\sqrt{2}cos(\frac{\pi}{4})+i\ sin(\frac{\pi}{4})]^6 [/Tex]
⇒ (2 + 2i)6 = [Tex](2\sqrt{2})^{6}[cos(\frac{6\pi}{4})+i\ sin(\frac{6\pi}{4})]\\ =(2\sqrt{2})^{6}[cos(\frac{3\pi}{2})+i\ sin(\frac{3\pi}{2})]\\ =(2\sqrt{2})^{6}[cos(\pi+\frac{\pi}{2})+i\ sin(\pi+\frac{\pi}{2})]\\ =(2\sqrt{2})^{6}[cos(\frac{\pi}{2})-i\ sin(\frac{\pi}{2})]\\ =(2\sqrt{2})^{6}[0-i]\\ =-(2\sqrt{2})^{6}i [/Tex]
⇒ (2 + 2i)6 = 512 (-i) = −512i
Example 3: Expand (1 + i)18.
Solution:
Here, r = [Tex]\sqrt{(1^2+1^2)} = \sqrt{2} [/Tex], θ = π/4
The polar form of (1+i) = [Tex][\sqrt{2}cos(\frac{\pi}{4})+i\ sin(\frac{\pi}{4})] [/Tex]
According to De Moivre’s Theorem: (cosθ + sinθ)n = cos(nθ) + isin(nθ).
Thus, (1 + i)18 = [Tex][\sqrt{2}cos(\frac{\pi}{4})+i\ sin(\frac{\pi}{4})]^{18} [/Tex]
⇒ (1 + i)18= [Tex](\sqrt{2})^{18}[cos(\frac{18\pi}{4})+i\ sin(\frac{18\pi}{4})]^{18}\\ =(\sqrt{2})^{18}[cos(\frac{9\pi}{2})+i\ sin(\frac{9\pi}{2})]\\ =(\sqrt{2})^{18}[cos(4\pi+\frac{\pi}{2})+i\ sin(4\pi+\frac{\pi}{2})]\\ =(\sqrt{2})^{18}[cos(\frac{\pi}{2})+i\ sin(\frac{\pi}{2})]\\ =(\sqrt{2})^{18}[0+i]\\ =(\sqrt{2})^{18}i [/Tex]
⇒ (1 + i)18 = 512i
Example 4: Expand (-√3 + 3i)31.
Solution:
Here, r = [Tex]\sqrt{((-\sqrt{3})^2+3^2)} = 2\sqrt{3} [/Tex], θ = 2π/3
The polar form of (-√3 + 3i) = [Tex][\sqrt{2}cos(\frac{\pi}{4})+i\ sin(\frac{\pi}{4})] [/Tex]
According to De Moivre’s Theorem: (cosθ + sinθ)n = cos(nθ) + i sin(nθ).
Thus, (-√3 + 3i)31= [Tex][2\sqrt{3}(cos(\frac{\pi}{4})+i\ sin(\frac{\pi}{4}))]^{31} = (2\sqrt{3})^{31}[cos(\frac{31\pi}{4})+i\ sin(\frac{31\pi}{4})]\\ =(2\sqrt{3})^{31}[cos(8\pi-\frac{\pi}{4})+i\ sin(8\pi-\frac{\pi}{4})]\\ =(2\sqrt{3})^{31}[cos(\frac{\pi}{4})-i\ sin(\frac{\pi}{4})]\\ =(2\sqrt{3})^{31}[\frac{1}{\sqrt{2}}-\frac{1}{\sqrt{2}}i] [/Tex]
Example 5: Expand (1 – i)10.
Solution:
r = [Tex]\sqrt{(1^2+(-1)^2)} = \sqrt{2} [/Tex], θ = π/4
The polar form of (1 – i) = [Tex]\sqrt{2}[cos(\frac{\pi}{4})+i \ sin(\frac{\pi}{4})] [/Tex]
According to De Moivre’s Theorem: (cosθ + sinθ)n = cos(nθ) + i sin(nθ).
Thus, (1 – i)10 = [Tex][\sqrt{2}cos(\frac{\pi}{4})+i\ sin(\frac{\pi}{4})]^{10} [/Tex]
= [Tex][\sqrt{2}(cos(\frac{Ï€}{4})+ i sin(\frac{Ï€}{4}))]^{10}\\ = (\sqrt2)^{10}[cos(\frac{10Ï€}{4})+i\ sin(\frac{10Ï€}{4})]\\ =(\sqrt2)^{10}[cos(\frac{5Ï€}{2})+i\ sin(\frac{5Ï€}{2})]\\ =(\sqrt2)^{10}[cos(2\pi+\frac{Ï€}{2})+i\ sin(2\pi+\frac{Ï€}{2})]\\ =(\sqrt2)^{10}[cos(\frac{Ï€}{2})-i\ sin(\frac{Ï€}{2})]\\ [/Tex]
= 32 [0 + i(-1)]
= 32 (-i)
= -32i
Example 6: Simplify (1 + √3i)6.
Solution:
Modulus of (1 + √3i)6 = [Tex]\sqrt{1^2+(\sqrt{3})^2} = 2 [/Tex]
Argument = tan-1(√3/1) = tan-1(√3) = π/3
⇒ Polar form = [Tex]2[cos(\frac{\pi}{3})+i\ sin(\frac{\pi}{3})] [/Tex]
Now, (1 + √3i)6 = [Tex][2(cos(\frac{\pi}{3})+i\ sin(\frac{\pi}{3}))]^6 [/Tex]
As per DeMoivre’s theorem, (cos x + isinx)n = cos(nx) + isin(nx).
⇒ [Tex][2(cos(\frac{\pi}{3})+i\ sin(\frac{\pi}{3}))]^6 [/Tex]
= [Tex]2^6(cos(\frac{6\pi}{3})+i\ sin(\frac{6\pi}{3})) [/Tex]
= 64 (cos 2Ï€ + i sin 2Ï€)
= 64(1 + 0)
= 64
Example 7: Simplify i√3.
Solution:
Modulus = r = [Tex]\sqrt{0^2+1^2} [/Tex] = 1
Argument = tan-1[1/0] = π/2
Polar Form = r[cosθ + isinθ] = [Tex]1[cos(\frac{\pi}{2}) +i\ sin(\frac{\pi}{2})] [/Tex]
Now, i^{√3} = [Tex][cos(\frac{\pi}{2}) + i\ sin(\frac{\pi}{2})]^{\sqrt3} [/Tex]
As per DeMoivre’s theorem: (cosθ + isinθ)n = cos(nθ) + isin(nθ).
⇒ [Tex][cos(\frac{\pi}{2}) + i\ sin(\frac{\pi}{2})]^{\sqrt3} [/Tex]
= [Tex][cos(\frac{\sqrt3\pi}{2}) + i\ sin(\frac{\sqrt3\pi}{2})]. [/Tex]
Complex Number Power Formula
Complex Numbers are numbers that can be written as a + ib, where a and b are real numbers and i (iota) is the imaginary component and its value is √(-1), and are often represented in rectangle or standard form. 10 + 5i, for example, is a complex number in which 10 represents the real component and 5i represents the imaginary part. Depending on the values of a and b, they might be wholly real or purely fictitious. When a = 0 in a + ib, ib is a totally imaginary number, and when b = 0, we get a, which is a strictly real number.
In this article, we will learn about, complex number power formulas, their examples, and others in detail.
Table of Content
- Complex Number Definition
- Complex Number Power Formula
- Complex Number Power Formula Derivation
- Solved Example on Complex Number Power Formula
- FAQs