Finding Complex Fourth Roots Exponential Form Step-by-Step Guide

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Introduction

In the realm of complex numbers, finding the roots of a complex number is a fundamental concept with applications in various fields such as electrical engineering, quantum mechanics, and signal processing. In this article, we will delve into the process of finding the complex fourth roots of a given complex number and expressing the roots in exponential form. Specifically, we will focus on the complex number $6 + 6\sqrt{3}i$. This comprehensive exploration will equip you with a solid understanding of how to tackle similar problems, enhancing your mathematical prowess and enabling you to apply this knowledge in diverse practical scenarios. Our journey begins with a foundational understanding of complex numbers and their representations, setting the stage for the more intricate calculations that follow.

Complex numbers, which extend the real number system by including the imaginary unit i (where i² = -1), are often expressed in the form a + bi, where a and b are real numbers. This is known as the rectangular form. However, complex numbers can also be represented in polar form, which is particularly useful when dealing with roots and powers of complex numbers. The polar form represents a complex number using its magnitude (or modulus) r and its argument (or angle) θ. The magnitude r is the distance from the origin to the point representing the complex number in the complex plane, while the argument θ is the angle formed by the line connecting the origin to the point and the positive real axis. This transformation from rectangular to polar form is crucial because it simplifies the process of finding roots and powers. Specifically, De Moivre's Theorem, a cornerstone of complex number theory, states that for any complex number in polar form r(cos θ + i sin θ) and any integer n, the n-th power of the complex number is rⁿ(cos nθ + i sin nθ). This theorem provides a direct method for raising complex numbers to integer powers and, by extension, for finding roots. The exponential form, another way to represent complex numbers, is closely related to the polar form. Using Euler's formula, which states that e^(iθ) = cos θ + i sin θ, we can express a complex number in exponential form as re^(iθ). This form is particularly convenient for performing multiplication, division, and finding roots of complex numbers, as it leverages the properties of exponential functions. The exponential form streamlines calculations and provides a compact notation, making it an indispensable tool in complex number manipulation. The process of finding the n-th roots of a complex number involves determining all complex numbers that, when raised to the power of n, yield the original complex number. This task becomes more manageable when complex numbers are expressed in polar or exponential form, as the roots are evenly spaced around a circle in the complex plane. Each root has the same magnitude (the n-th root of the original magnitude) but different arguments. These arguments are determined by dividing the argument of the original complex number by n and adding multiples of 2π/n. This geometric interpretation of complex roots is fundamental to understanding their distribution and properties. By mastering these concepts, you not only gain a deeper appreciation for complex numbers but also acquire powerful tools for solving a wide range of mathematical and engineering problems.

Converting to Polar Form

Before we can find the fourth roots, we need to convert the given complex number, $6 + 6\sqrt{3}i$, into polar form. Polar form expresses a complex number in terms of its magnitude (r) and argument (θ). The magnitude represents the distance from the origin to the point representing the complex number in the complex plane, while the argument is the angle formed by the line connecting the origin to the point and the positive real axis. This conversion is essential because polar form simplifies the process of finding roots and powers of complex numbers. To convert a complex number from its rectangular form (a + bi) to polar form (r(cos θ + i sin θ)), we first calculate the magnitude r. The magnitude r is given by the formula r = √(a² + b²). In our case, a = 6 and b = 6√3, so we have:

r = √(6² + (6√3)²) = √(36 + 108) = √144 = 12

Thus, the magnitude of the complex number is 12. Next, we need to find the argument θ. The argument θ is the angle whose tangent is the ratio of the imaginary part to the real part, i.e., tan θ = b/a. In our case, tan θ = (6√3)/6 = √3. To find the principal value of θ, we consider the quadrant in which the complex number lies in the complex plane. Since both the real and imaginary parts are positive, the complex number lies in the first quadrant. The angle whose tangent is √3 in the first quadrant is π/3. Therefore, the principal argument θ is π/3. Knowing the magnitude and argument, we can express the complex number in polar form. The polar form of $6 + 6\sqrt{3}i$ is:

12(cos(π/3) + i sin(π/3))

This representation is crucial for finding the roots because it allows us to use De Moivre's Theorem, which provides a direct method for calculating powers and roots of complex numbers. De Moivre's Theorem states that for any complex number in polar form r(cos θ + i sin θ) and any integer n, the n-th power of the complex number is rⁿ(cos nθ + i sin nθ). By converting to polar form, we can leverage this theorem to simplify the process of finding the fourth roots. The ability to convert between rectangular and polar forms is a fundamental skill in complex number theory. It not only simplifies calculations but also provides a deeper understanding of the geometric representation of complex numbers. The polar form allows us to visualize complex numbers as points in the complex plane, with the magnitude representing the distance from the origin and the argument representing the angle with the positive real axis. This geometric perspective is invaluable for solving complex number problems and understanding their applications in various fields. In summary, converting the complex number $6 + 6\sqrt{3}i$ to polar form involves finding its magnitude and argument, which allows us to express it as 12(cos(π/3) + i sin(π/3)). This step is essential for applying De Moivre's Theorem and finding the complex fourth roots.

Applying De Moivre's Theorem

Now that we have the complex number in polar form, 12(cos(π/3) + i sin(π/3)), we can proceed to find its fourth roots using De Moivre's Theorem. De Moivre's Theorem is a fundamental concept in complex number theory, providing a powerful tool for calculating powers and roots of complex numbers. It states that for any complex number in polar form r(cos θ + i sin θ) and any integer n, the n-th power of the complex number is rⁿ(cos nθ + i sin nθ). However, when finding the n-th roots of a complex number, we need to consider all possible roots, which are evenly spaced around a circle in the complex plane. This requires a slight modification of the theorem to account for the periodic nature of trigonometric functions. To find the fourth roots of 12(cos(π/3) + i sin(π/3)), we first take the fourth root of the magnitude. The magnitude of our complex number is 12, so the fourth root of the magnitude is 12^(1/4), which can also be written as the fourth root of 12 (∜12). This value will be the magnitude of each of the four complex fourth roots. Next, we need to find the arguments of the fourth roots. The argument of the original complex number is π/3. To find the arguments of the fourth roots, we divide the original argument by 4 and add multiples of 2π/4 (which simplifies to π/2) to account for all possible roots. This is because there are four distinct fourth roots, evenly spaced around the complex plane. The general formula for the arguments of the n-th roots is (θ + 2πk)/ n, where k is an integer ranging from 0 to n - 1. In our case, n = 4, and the original argument θ is π/3. So, the arguments of the fourth roots are given by:

(π/3 + 2πk)/4, for k = 0, 1, 2, 3

Let's calculate these arguments:

  • For k = 0: (π/3 + 2π(0))/4 = π/12
  • For k = 1: (π/3 + 2π(1))/4 = (π/3 + 2π)/4 = (7π/3)/4 = 7π/12
  • For k = 2: (π/3 + 2π(2))/4 = (π/3 + 4π)/4 = (13π/3)/4 = 13π/12
  • For k = 3: (π/3 + 2π(3))/4 = (π/3 + 6π)/4 = (19π/3)/4 = 19π/12

These arguments correspond to the four complex fourth roots. Now we can express each root in polar form. The four complex fourth roots are:

  • ∜12(cos(π/12) + i sin(π/12))
  • ∜12(cos(7π/12) + i sin(7π/12))
  • ∜12(cos(13π/12) + i sin(13π/12))
  • ∜12(cos(19π/12) + i sin(19π/12))

This application of De Moivre's Theorem allows us to systematically find all the roots of a complex number, which is a crucial skill in complex analysis. The roots are evenly distributed around a circle in the complex plane, reflecting the periodic nature of complex exponentials. By understanding and applying De Moivre's Theorem, we can solve a wide range of problems involving complex roots and powers.

Expressing in Exponential Form

Having found the fourth roots in polar form, the final step is to express them in exponential form. The exponential form of a complex number is a compact and convenient representation that leverages Euler's formula, which states that e^(iθ) = cos θ + i sin θ. This formula provides a direct link between the trigonometric representation (polar form) and the exponential representation of a complex number. Expressing complex numbers in exponential form simplifies many mathematical operations, especially multiplication, division, and finding powers and roots. The general form of a complex number in exponential form is re^(iθ), where r is the magnitude and θ is the argument of the complex number. In our case, we have already found the magnitudes and arguments of the four complex fourth roots. The magnitude for all four roots is ∜12, and the arguments are π/12, 7π/12, 13π/12, and 19π/12. Now, we can simply plug these values into the exponential form re^(iθ) to obtain the exponential representations of the roots. The first root has magnitude ∜12 and argument π/12, so its exponential form is:

∜12 e^(iπ/12)

The second root has magnitude ∜12 and argument 7π/12, so its exponential form is:

∜12 e^(i7π/12)

The third root has magnitude ∜12 and argument 13π/12, so its exponential form is:

∜12 e^(i13π/12)

The fourth root has magnitude ∜12 and argument 19π/12, so its exponential form is:

∜12 e^(i19π/12)

These are the four complex fourth roots of $6 + 6\sqrt{3}i$ expressed in exponential form. The exponential form is particularly useful for visualizing the roots in the complex plane. Each root lies on a circle with radius ∜12, and the arguments determine their angular positions. The roots are evenly spaced around the circle, with an angular separation of π/2 (since we are finding the fourth roots). This geometric interpretation is a direct consequence of De Moivre's Theorem and the properties of complex exponentials. Expressing complex numbers in exponential form not only provides a compact notation but also facilitates algebraic manipulations. For example, multiplying two complex numbers in exponential form involves multiplying their magnitudes and adding their arguments, which is often simpler than performing the same operation in rectangular form. Similarly, raising a complex number in exponential form to a power involves raising the magnitude to that power and multiplying the argument by the power. In summary, converting the fourth roots from polar form to exponential form involves applying Euler's formula and expressing each root as re^(iθ), where r is the magnitude and θ is the argument. This representation provides a concise and mathematically convenient way to express complex numbers and is particularly useful for advanced mathematical and engineering applications. By mastering this conversion, you gain a valuable tool for working with complex numbers and solving a wide range of problems.

Conclusion

In this comprehensive exploration, we have successfully found the complex fourth roots of $6 + 6\sqrt{3}i$ and expressed them in exponential form. This process involved several key steps, each building upon the previous one to achieve the final result. First, we converted the complex number from its rectangular form to polar form, which is essential for applying De Moivre's Theorem. This conversion required calculating the magnitude and argument of the complex number, providing a geometric interpretation of its position in the complex plane. Next, we applied De Moivre's Theorem to find the four distinct fourth roots. This involved taking the fourth root of the magnitude and dividing the argument by 4, while also accounting for the periodic nature of complex roots by adding multiples of 2π/4. This step highlights the power and elegance of De Moivre's Theorem in simplifying the process of finding roots of complex numbers. Finally, we expressed the roots in exponential form using Euler's formula. This representation provides a compact and mathematically convenient way to write complex numbers, facilitating further calculations and analysis. The exponential form also allows for a clear visualization of the roots in the complex plane, as they are evenly spaced around a circle with a radius equal to the fourth root of the original magnitude. Throughout this process, we have emphasized the importance of understanding the underlying concepts and techniques in complex number theory. The ability to convert between rectangular, polar, and exponential forms is crucial for solving a wide range of problems in mathematics, engineering, and physics. Furthermore, the application of De Moivre's Theorem is a fundamental skill for finding powers and roots of complex numbers. By mastering these techniques, you gain a deeper appreciation for the structure and properties of complex numbers, which are essential tools in many areas of science and technology. The process of finding complex roots is not only a mathematical exercise but also a gateway to understanding more advanced topics such as complex analysis, Fourier analysis, and quantum mechanics. These fields rely heavily on the properties of complex numbers and their roots, making this knowledge invaluable for further study and research. In conclusion, the four complex fourth roots of $6 + 6\sqrt{3}i$ in exponential form are:

  • ∜12 e^(iπ/12)
  • ∜12 e^(i7π/12)
  • ∜12 e^(i13π/12)
  • ∜12 e^(i19π/12)

This result demonstrates the power of complex number theory and the elegance of exponential representation. By mastering these concepts, you are well-equipped to tackle a variety of problems involving complex numbers and their applications.