Imaginary 6th Root Calculator

This calculator finds the imaginary 6th root of complex numbers. The imaginary 6th root is one of the six complex numbers that, when raised to the 6th power, equal the original complex number.

What is the Imaginary 6th Root?

The imaginary 6th root of a complex number z = a + bi is a complex number w = c + di such that w⁶ = z. Complex numbers have six roots in total, including both real and imaginary roots.

For a complex number z = a + bi, the six roots can be calculated using polar form and De Moivre's Theorem. The principal root is typically chosen as the one with the smallest positive argument.

How to Calculate the Imaginary 6th Root

To find the imaginary 6th root of a complex number:

Convert the complex number to polar form (r, θ)
Calculate the sixth root of the magnitude: r^(1/6)
Divide the angle by 6: θ/6
Add 2πk/6 for each root (k = 0 to 5)
Convert back to rectangular form using Euler's formula

The imaginary roots will have non-zero imaginary parts, while the real roots will have zero imaginary parts.

The Formula

The general formula for the k-th imaginary 6th root of a complex number z = a + bi is:

w_k = (r)^1/6 * [cos((θ + 2πk)/6) + i sin((θ + 2πk)/6)]
where r = √(a² + b²), θ = arctan(b/a), and k = 1, 2, 3, 4, 5

For the principal root (k=0), the formula becomes:

w₀ = (r)^1/6 * [cos(θ/6) + i sin(θ/6)]

Worked Example

Let's find the imaginary 6th roots of the complex number 8 + 8i.

Convert to polar form:
- r = √(8² + 8²) = √(64 + 64) = √128 ≈ 11.3137
- θ = arctan(8/8) = π/4 radians (45°)
Calculate the magnitude root: r^(1/6) ≈ 11.3137^(1/6) ≈ 1.6
Calculate the angle for the first imaginary root (k=1):
- θ/6 = π/24 ≈ 0.1309 radians
- Add 2π/6 = π/3 ≈ 1.0472 radians
- Total angle = π/24 + π/3 ≈ 1.1781 radians
Convert to rectangular form:
- Real part: 1.6 * cos(1.1781) ≈ 1.6 * 0.411 ≈ 0.658
- Imaginary part: 1.6 * sin(1.1781) ≈ 1.6 * 0.911 ≈ 1.458

The first imaginary root is approximately 0.658 + 1.458i.

FAQ

What is the difference between real and imaginary roots?

Real roots have zero imaginary parts, while imaginary roots have non-zero imaginary parts. The imaginary roots of a complex number are the ones with non-zero imaginary components.

How many imaginary roots does a complex number have?

A complex number has five imaginary roots (excluding the real roots). Together with the real roots, there are six roots in total for any non-zero complex number.

Can I find the imaginary roots of a real number?

Yes, real numbers are a subset of complex numbers with zero imaginary parts. The imaginary roots of a real number will have non-zero imaginary parts.