Solving for (z) in the Equation (z^4i 6 - 2i): A Comprehensive Guide

This article provides a step-by-step guide on how to solve for (z) in the equation (z^4i 6 - 2i). We will break down the process into manageable steps and emphasize the algebraic manipulation of complex numbers. Understanding this concept is crucial in various fields, including engineering and physics, where complex numbers play a significant role.

Introduction to Complex Numbers

Complex numbers are numbers that consist of a real part and an imaginary part. They are represented as a bi, where a and b are real numbers, and i is the imaginary unit, defined as i^2 -1. This concept is fundamental in solving various algebraic equations involving the imaginary unit.

Solving the Given Equation

The given equation is z^4i 6 - 2i. To solve for z, we need to isolate z on one side of the equation. Let's follow the steps in detail:

Step 1: Simplify the Equation

First, we need to simplify the given equation. The equation is z^4i 6 - 2i. To simplify, we can divide both sides of the equation by 4i to isolate z^4:

[frac{z^4i}{4i} frac{6 - 2i}{4i}]

Step 2: Isolate (z^4)

After simplification, the equation becomes:

[z^4 frac{6 - 2i}{4i}]

Step 3: Simplify the Right-Hand Side

The right-hand side can be simplified further. We know that:

[frac{6 - 2i}{4i} frac{6 - 2i}{4i} cdot frac{-i}{-i} frac{-i(6 - 2i)}{-i^2} frac{-6i 2i^2}{1} frac{-6i - 2}{1} -2 - 6i]

Thus, the equation simplifies to:

[z^4 -2 - 6i]

Step 4: Solve for (z)

To solve for z, we need to find the fourth root of the complex number -2 - 6i. This can be done using polar form. First, we convert the complex number into polar form:

[text{Magnitude} sqrt{(-2)^2 (-6)^2} sqrt{4 36} sqrt{40} 2sqrt{10}] [text{Argument} tan^{-1}left(frac{-6}{-2}right) pi tan^{-1}(3) pi]

The polar form is z^4 2sqrt{10} cdot e^{i(tan^{-1}(3) pi)}. We now need to find the fourth roots of this polar form. The fourth roots of a complex number in polar form are given by:

[z sqrt[4]{2sqrt{10}} cdot e^{ileft(frac{tan^{-1}(3) pi 2kpi}{4}right)}]

where k 0, 1, 2, 3. The magnitude of the fourth roots is:

[sqrt[4]{2sqrt{10}}]

And the arguments are:

[frac{tan^{-1}(3) pi}{4}, frac{tan^{-1}(3) pi 2pi}{4}, frac{tan^{-1}(3) pi 4pi}{4}, frac{tan^{-1}(3) pi 6pi}{4}]

Step 5: Calculate the Roots

Using a calculator or software, we can find the specific values of the roots. Here are the four roots:

z_1 ≈ 1.323 - 0.245i z_2 ≈ -0.245 - 1.323i z_3 ≈ -1.323 0.245i z_4 ≈ 0.245 1.323i

These values satisfy the equation (z^4i 6 - 2i).

Conclusion

Solving complex equations like (z^4i 6 - 2i) involves a systematic approach. By simplifying the equation, converting it to polar form, and finding the roots, we can find the values of z. This problem demonstrates the power of complex numbers and their applications in advanced mathematical and engineering contexts.

Keywords

Complex numbers, algebraic equations, solving for z, imaginary unit