Solving ##z^3 + 8 for z = 1 + i*sqrt(3)##

Algebraic Formulation of the Complex Root

The fundamental problem involves the evaluation of a specific complex expression given a defined root. We start with the primary variable ##z##, which is expressed in standard rectangular form. In this context, the real component is denoted by ##x = 1## and the imaginary component is denoted by ##y = \sqrt{3}##. The objective is to compute the value of the cubic sum ##z^3 + 8##. This requires a rigorous application of algebraic principles and the manipulation of the imaginary unit ##i##, where ##i^2 = -1##.

To begin our technical analysis, we must acknowledge the constraints provided in the problem statement. The expression ##z = 1 + i\sqrt{3}## represents a point in the complex plane that possesses specific magnitude and directional properties. By treating this variable as a binomial, we can proceed with various methods of evaluation, including direct substitution or transformation into alternative mathematical systems. The precise nature of the coefficients suggests that this root is related to specific geometric rotations. We will evaluate the expression using the following method “” in our analysis to ensure numerical accuracy.

The complexity of the calculation depends on the preservation of the imaginary unit throughout the power operation. In many technical scenarios, complex numbers like ##z## appear as solutions to polynomial equations or as representations of phase in oscillating systems. By defining the input as ##1 + i\sqrt{3}##, we are working with a number whose components are integers and irrational square roots. This structure allows for significant simplification when raised to integer powers, particularly powers that are multiples of three. Understanding these properties is essential before performing any operations on the cubic expression.

Geometric Interpretation on the Argand Plane

Visualizing the complex number on an Argand diagram provides a technical intuition for the subsequent calculations. The point ##(1, \sqrt{3})## is located in the first quadrant of the complex plane, where both the real and imaginary parts are positive. The distance from the origin to this point, known as the modulus, is a critical variable in our evaluation. By plotting this coordinate, we can observe that it forms a right-angled triangle with the real axis, which simplifies the determination of its angular displacement.

The angular displacement, or argument, is calculated using the inverse tangent of the ratio between the imaginary and real components. Specifically, we evaluate ##\theta = \arctan(\sqrt{3}/1)##, which yields a precise value of ##\pi/3## radians or sixty degrees. This specific angle is significant because it is a rational fraction of a full rotation. In the complex plane, multiplication by such a number corresponds to a specific rotation and scaling. Recognizing this geometric configuration helps us predict that raising the number to the third power will result in a purely real outcome.

Furthermore, the modulus ##r## is determined through the Pythagorean theorem applied to the components. We calculate ##r = \sqrt{1^2 + (\sqrt{3})^2} = \sqrt{4} = 2##. Thus, the complex number ##z## can be viewed as a vector of length two pointing in a direction of sixty degrees from the positive real axis. When we square or cube this vector, the length is scaled exponentially while the angle is multiplied linearly. This geometric property is the foundation of complex power evaluation and serves as a cross-verification for the algebraic methods used later.

Step-by-Step Analytical Derivation

Transformation to Polar and Exponential Forms

The most efficient way to evaluate powers of complex numbers is to transition from rectangular form to polar or exponential form. Using the previously derived modulus ##r = 2## and argument ##\theta = \pi/3##, we can rewrite the variable. In polar form, the expression becomes ##z = 2(\cos(\pi/3) + i \sin(\pi/3))##. This representation explicitly separates the magnitude of the number from its rotational component, which is vital for the application of power rules. The precision of the trigonometric values ensures the integrity of the final result.

Transitioning further into the exponential form using Euler’s formula provides a more compact notation for calculation. We express the variable as ##z = 2e^{i\pi/3}##. In this format, the base of the natural logarithm ##e## is raised to an imaginary power that signifies the rotation in radians. This approach is preferred in engineering and physics because it simplifies the laws of exponents. The transition to exponential form allows us to handle the cubic power by simply applying the power to the coefficient and the exponent independently.

Mathematically, the conversion process is defined by the following equations:

r = sqrt(Re(z)^2 + Im(z)^2)
theta = atan2(Im(z), Re(z))
z = r * exp(i * theta)

Substituting our values, we confirm that ##r = 2## and ##\theta = \pi/3##. This set of specifications establishes the groundwork for the next phase of the derivation. By maintaining technical rigor in these early steps, we eliminate the risk of cumulative errors in the cubic expansion. The exponential form is inherently more robust for high-order power operations compared to binomial expansion.

Application of De Moivre’s Theorem

De Moivre’s Theorem states that for any complex number in polar form and any integer ##n##, the power is given by raising the modulus to ##n## and multiplying the argument by ##n##. Applying this to our cubic expression, we find that ##z^3 = (2e^{i\pi/3})^3##. Following the algebraic rules for exponents, this simplifies to ##2^3 \cdot e^{i(\pi/3 \cdot 3)}##. The calculation of the modulus component is straightforward, as ##2^3 = 8##, which sets the magnitude for the resulting complex vector.

The rotation component of the calculation is where the simplification occurs. Multiplying the argument ##\pi/3## by the exponent ##3## results in a final argument of ##\pi## radians. In the complex plane, an angle of ##\pi## corresponds to a direction pointing directly along the negative real axis. Therefore, the resulting complex number ##z^3## will have no imaginary component. The theorem allows us to skip the tedious step of expanding binomial terms and directly identify the position of the cubed value in the complex field.

To find the numerical value, we convert back to rectangular form using the trigonometric components of the new argument. We have:

###z^3 = 8(\cos(\pi) + i \sin(\pi))###

Since ##\cos(\pi) = -1## and ##\sin(\pi) = 0##, the expression simplifies to ##8(-1 + 0i) = -8##. This result is a purely real number, which is a common characteristic of powers of roots of unity scaled by a constant. Having found that ##z^3 = -8##, the final step of adding eight to this value is trivial, leading to a sum of zero.

Verification via Binomial Expansion and Rectangular Arithmetic

Expansion of the Cubic Binomial Expression

While the exponential method is technically superior for efficiency, verifying the result through binomial expansion ensures that no conceptual errors were made during the transformation. We treat ##z = 1 + i\sqrt{3}## as a binomial ##(a + b)## where ##a = 1## and ##b = i\sqrt{3}##. The expansion follows the standard identity ###(a + b)^3 = a^3 + 3a^2b + 3ab^2 + b^3###. This approach requires careful management of the imaginary unit and the coefficients to maintain accuracy throughout the multi-step arithmetic process.

Substituting the specific values into the identity yields the following expression:

###z^3 = (1)^3 + 3(1)^2(i\sqrt{3}) + 3(1)(i\sqrt{3})^2 + (i\sqrt{3})^3###

Each term must be evaluated individually to observe how the real and imaginary parts interact. The first term is simply one. The second term involves a single power of ##i##, resulting in ##3i\sqrt{3}##. The third term involves the square of ##i\sqrt{3}##, which translates to ##3(-3) = -9## because ##i^2 = -1## and the square of the root is three. This systematic breakdown is essential for a technical verification.

The final term in the expansion is the cube of ##i\sqrt{3}##. This is calculated as ##i^3 \cdot (\sqrt{3})^3##. We know that ##i^3 = -i## and ##(\sqrt{3})^3 = 3\sqrt{3}##, so the fourth term becomes ##-3i\sqrt{3}##. By expanding the expression in this manner, we can clearly see the components that will eventually cancel each other out. This method provides a transparent view of the internal mechanics of complex multiplication that polar forms often obscure, making it a valuable tool for validation in rigorous mathematical proofs.

Algebraic Simplification of Real and Imaginary Parts

With all terms expanded, we now group the real and imaginary components to find the simplified value of ##z^3##. The full expanded expression is ##1 + 3i\sqrt{3} – 9 – 3i\sqrt{3}##. It is immediately apparent that the terms containing the imaginary unit ##i## are identical in magnitude but opposite in sign. Specifically, ##3i\sqrt{3}## and ##-3i\sqrt{3}## sum to zero, effectively removing the imaginary dimension from the final result. This cancellation confirms that the cube of our original number is a purely real integer.

The remaining real components are ##1## and ##-9##. Subtracting nine from one results in ##-8##. This matches the result obtained through De Moivre’s Theorem, providing high confidence in the accuracy of the derivation. The consistency between the geometric rotation method and the algebraic expansion method is a hallmark of complex number theory. At this stage, we have established with mathematical certainty that the value of ##z^3## is precisely negative eight. This concludes the most intensive part of the technical evaluation of the given variable.

The final calculation involves the addition of the constant specified in the original problem statement. Given that we have derived ##z^3 = -8##, the expression ##z^3 + 8## becomes ##-8 + 8##. The sum of these two values is zero. In a professional context, identifying that a complex expression evaluates to zero often indicates that the original number is a root of the polynomial equation ##x^3 + 8 = 0##. This verification step completes the analytical process and allows us to transition into the broader theoretical implications of this specific problem.

Theoretical Deep Dive and Mathematical Context

Relationship to the Roots of Unity

The number ##z = 1 + i\sqrt{3}## is not an arbitrary complex value; it is deeply connected to the theory of the roots of unity. If we divide ##z## by its modulus, we get ##w = 1/2 + i\sqrt{3}/2##. This value is a primitive sixth root of unity, often denoted as ##\omega_6##. A primitive ##n##-th root of unity is a complex number that, when raised to the power of ##n##, equals one, and no smaller positive power yields one. Understanding this context helps explain why the powers of ##z## behave in such a predictable and periodic manner.

Because ##z/2## is a sixth root of unity, we can state that ##(z/2)^6 = 1##. Consequently, ##(z/2)^3## must be a square root of one. Since the argument of ##z/2## is ##\pi/3##, the argument of its cube is ##\pi##, which corresponds to ##-1##. This implies that ##z^3 / 2^3 = -1##, leading directly back to the conclusion that ##z^3 = -8##. The study of roots of unity is a cornerstone of modern algebra, particularly in fields like Galois theory and cryptography, where cyclic structures are utilized for security and error correction.

In the context of the unit circle, these roots form the vertices of regular polygons. The number ##z##, after normalization, represents one vertex of a regular hexagon centered at the origin. When we cube this value, we are effectively skipping vertices in a systematic rotation that eventually lands on the negative real axis. This behavior is a specific instance of the broader principle that the ##n##-th roots of a number ##A## are distributed evenly at angles of ##2\pi/n## around a circle of radius ##A^{1/n}##. Such technical insights are vital for advanced mathematical modeling.

Polynomial Factorization and General Solutions

The problem of evaluating ##z^3 + 8## can also be framed as a search for the roots of the cubic polynomial ##P(x) = x^3 + 8##. Using the sum of cubes factorization, we know that ###x^3 + a^3 = (x + a)(x^2 – ax + a^2)###. For our specific case where ##a = 2##, the factorization becomes ##(x + 2)(x^2 – 2x + 4)##. Setting this equal to zero allows us to find all three roots of the equation. One root is obviously the real number ##x = -2##, which we encountered in the argument of ##\pi## earlier.

The other two roots are found by solving the quadratic factor ##x^2 – 2x + 4 = 0## using the quadratic formula. Applying the formula, we get ##x = [2 \pm \sqrt{(-2)^2 – 4(1)(4)}] / 2##, which simplifies to ##(2 \pm \sqrt{-12}) / 2##. Further simplification yields ##(2 \pm 2i\sqrt{3}) / 2##, or ##1 \pm i\sqrt{3}##. This confirms that our original value ##z = 1 + i\sqrt{3}## is exactly one of the three complex roots of the equation ##x^3 + 8 = 0##. This relationship explains why the expression evaluated to zero.

Historically, the study of cubic equations led to the development of complex numbers themselves. Mathematicians like Cardan and Bombelli realized that even when roots are real, the intermediate steps of solving cubic formulas often require the square roots of negative numbers. Our evaluation of ##z^3 + 8## is a modern reflection of these foundational discoveries. By understanding the polynomial context, we see that the problem is not just a calculation but a demonstration of the Fundamental Theorem of Algebra, which guarantees that a cubic polynomial will always have exactly three roots in the complex plane.