91Ó°ÊÓ

To solve the equation \(\text{x}^4 = 16\) in the complex number system, we need to consider the concept of complex roots of unity. Complex roots of unity are the solutions to the equation \(x^n = 1\) where \(n\) is a positive integer. These roots are equally spaced on the unit circle in the complex plane.
For any \(\text{n-th}\) root of unity, we use the formula \(e^{i \frac{2k\text{Ï€}}{n}}\) where \(k\) ranges from 0 to \(n-1\). In our problem, we need the fourth roots of unity. The fourth roots of unity are therefore given by \(\text{e}^{i \frac{2k\text{Ï€}}{4}}\) for \(k = 0, 1, 2 \text{ and } 3\).
In simpler terms, these represent angles of \(0, \frac{\text{Ï€}}{2}, \text{Ï€}, \text{ and } \frac{3\text{Ï€}}{2} \) radians on the unit circle:

• When \(k = 0\): \(e^{i \times 0} = 1\)
• When \(k = 1\): \(e^{i \frac{\text{Ï€}}{2}} = i \)
• When \(k = 2\): \(e^{i \text{Ï€}} = -1 \)
• When \(k = 3\): \(e^{i \frac{3\text{Ï€}}{2}} = -i \)

These roots help us fully solve the original equation by taking into account all possible solutions within the complex plane.

A fourth root of a complex number is one of the four numbers that, when raised to the power of four, yield the initial number.
In our equation \(x^4 = 16\), solving for \(x\) means finding all fourth roots of 16. To do this, we first express 16 in the form of \(2^4\), then use the fourth roots of unity:
\[\begin{equation} x = 2 e^{i\frac{2k\text{Ï€}}{4}} \end{equation}\]. We follow these steps:

• First, take the principal root: \(x = 2 \)
• Then, we apply the roots of unity: multiply by \(e^{i\frac{2k\text{Ï€}}{4}}\) for \(k=0, 1, 2, \text{ and } 3\)

For \_k = 0\_, \_x = 2 e^{i \times 0} = 2\_. For \_k = 1\_, \_x = 2 e^{i \frac{\text{Ï€}}{2}} = 2i\_. For \_k = 2\_, \_x = 2 e^{i \text{Ï€}} = -2\_. For \_k = 3\_, \_x = 2 e^{i \frac{3\text{Ï€}}{2}} = -2i\_.
::The fourth roots of 16 are therefore: \(+2, 2i, -2, \text{ and } -2i\).

Solving equations in the complex number system involves considering both real and imaginary parts. Complex numbers are of the form \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part.
In this system, a number like \(16\) can take on several roots, both real and imaginary.

• The equations can often be rewritten or reinterpreted using Euler's formula: \(e^{i\theta} = \text{cos}(\theta) + i \text{sin}(\theta)\), which is critical for understanding rotations in the complex plane.
• When raising to fractions of \(\text{Ï€}\), such as \(e^{i\frac{\text{Ï€}}{2}}\), the imaginary unit \(i\), an undefined square root appears.

This means that for any equation of the form \(x^n = a\), we look at all n possible roots in the complex number system. These roots exhibit symmetry along the unit circle, and include all combinations of real and imaginary solutions. Understanding this system allows us to solve equations like \(x^4 = 16\) completely, providing all possible roots and ensuring no solutions are overlooked.