Complex Number Exponents Calculator

Complex Number Exponents

What are Complex Number Exponents?

Complex number exponents involve raising a complex number to a power. This operation combines concepts from complex numbers and exponentiation, resulting in a new complex number.

Formula

The formula for complex number exponents uses De Moivre's theorem:

\[ (r(\cos\theta + i\sin\theta))^n = r^n(\cos(n\theta) + i\sin(n\theta)) \]

Where:

• \(r\) is the magnitude of the complex number
• \(\theta\) is the argument (angle) of the complex number
• \(n\) is the exponent
• \(i\) is the imaginary unit (\(\sqrt{-1}\))

Calculation Steps

Let's calculate (2 + 3i)^4:

1. Express in polar form:
   \(r = \sqrt{2^2 + 3^2} = \sqrt{13}\)
   \(\theta = \tan^{-1}(\frac{3}{2}) \approx 0.9828\)
2. Apply De Moivre's formula:
   \((\sqrt{13}(\cos 0.9828 + i\sin 0.9828))^4\)
3. Calculate:
   \(r^4 = (\sqrt{13})^4 = 13^2 = 169\)
   \(4\theta = 4 \times 0.9828 \approx 3.9312\)
4. Result:
   \(169(\cos 3.9312 + i\sin 3.9312)\)
   \(\approx -119 - 120i\)

Example and Visual Representation

Let's visualize (2 + 3i)^4:

This visual representation shows:

• The original complex number 2+3i in blue
• The result (2+3i)^4 ≈ -119-120i in red
• The original angle θ in green
• The new angle 4θ in orange