Roots of Complex Numbers Calculator

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Example

Created on 2024-06-20Asked by Luna Martinez (Solvelet student)

Find all the cube roots of the complex number

z = 8 + 8i

Solution

To find all the cube roots of the complex number

z = 8 + 8i

: 1. **Express

z

in polar form:**

z = 8 + 8i = 8\sqrt{2}\left(\cos\frac{\pi}{4} + i\sin\frac{\pi}{4}\right).

2. **Apply De Moivre's Theorem for cube roots:**

z^{1/3} = \left(8\sqrt{2}\right)^{1/3} \left(\cos\frac{\pi/4 + 2k\pi}{3} + i\sin\frac{\pi/4 + 2k\pi}{3}\right), \quad k = 0, 1, 2.

3. **Calculate the cube roots:**

\left(8\sqrt{2}\right)^{1/3} = 2\sqrt[3]{2}.

For

k = 0

\cos\frac{\pi/12} + i\sin\frac{\pi/12}.

For

k = 1

\cos\frac{9\pi/12} + i\sin\frac{9\pi/12}.

For

k = 2

\cos\frac{17\pi/12} + i\sin\frac{17\pi/12}.

4. **Result:** The three cube roots are:

2\sqrt[3]{2}\left(\cos\frac{\pi}{12} + i\sin\frac{\pi}{12}\right),

2\sqrt[3]{2}\left(\cos\frac{3\pi}{4} + i\sin\frac{3\pi}{4}\right),

2\sqrt[3]{2}\left(\cos\frac{17\pi}{12} + i\sin\frac{17\pi}{12}\right).

Solved on Solvelet with Basic AI Model

Some of the related questions asked by Alexander Sanchez on Solvelet

1. Find all the roots of the equation

z^3 = 8

in the complex plane.2. Determine the roots of the equation

z^4 = -16

in the complex plane.,

DefinitionThe roots of a complex number are the solutions to equations of the form zn=w where z, w are complex numbers. The roots can be obtained using De Moivre's method as follows (z_k = Solution Roots of Complex numbers Finding using De moivre's Formula.

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