Simplify Complex Number: $(-6+6i)^2$

by Alex Johnson 37 views

Let's dive into simplifying the complex number expression $(-6+6i)^2$. This is a common task in mathematics, especially when working with complex numbers in algebra and calculus. Simplifying expressions like this helps us understand the properties of complex numbers and how they behave under operations such as exponentiation. When we encounter a problem like $(-6+6i)^2$, we're essentially being asked to multiply the complex number $(-6+6i)$ by itself. This process involves applying the distributive property, much like we would with binomials in real number algebra. The key difference here is the presence of the imaginary unit, $i$, which has a special property: $i^2 = -1$. This property is crucial for simplifying the final result. So, the first step is to expand the expression. We can think of $(-6+6i)^2$ as $(-6+6i) imes (-6+6i)$. Now, we use the FOIL method (First, Outer, Inner, Last) or simply distribute each term of the first binomial to each term of the second. Let's go through it:

  • First terms: $(-6) imes (-6) = 36$.
  • Outer terms: $(-6) imes (6i) = -36i$.
  • Inner terms: $(6i) imes (-6) = -36i$.
  • Last terms: $(6i) imes (6i) = 36i^2$.

Now, we combine these results: $36 - 36i - 36i + 36i^2$. Notice the $36i^2$ term. This is where we use the property $i^2 = -1$. Substituting $-1$ for $i^2$, we get $36(-1) = -36$. So, our expression becomes $36 - 36i - 36i - 36$. The next step is to combine like terms. We group the real parts (numbers without $i$) and the imaginary parts (numbers with $i$). The real parts are $36$ and $-36$. Adding them gives $36 + (-36) = 0$. The imaginary parts are $-36i$ and $-36i$. Adding them gives $-36i + (-36i) = -72i$. Therefore, the simplified form of $(-6+6i)^2$ is $0 - 72i$, which is simply $-72i$. This process demonstrates how to handle the squaring of a complex number, a fundamental skill in understanding more advanced complex number operations and their applications in various fields of science and engineering.

Understanding Complex Numbers and Exponentiation

Working with complex numbers, especially when they are raised to a power, can seem daunting at first. However, by breaking down the process and understanding the fundamental properties of complex numbers, it becomes much more manageable. The expression $(-6+6i)^2$ involves squaring a complex number, which is a specific case of exponentiation. A complex number is generally expressed in the form $a + bi$, where $a$ is the real part and $b$ is the imaginary part, and $i$ is the imaginary unit satisfying $i^2 = -1$. When we square such a number, $(a+bi)^2$, we are essentially applying the binomial expansion formula: $(a+b)^2 = a^2 + 2ab + b^2$. In our case, $a = -6$ and $b = 6i$. So, applying this formula, we get:

  • $a^2 = (-6)^2 = 36$.
  • $2ab = 2 imes (-6) imes (6i) = -72i$.
  • $b^2 = (6i)^2 = 6^2 imes i^2 = 36 imes i^2$.

Again, we use the property $i^2 = -1$. So, $b^2 = 36 imes (-1) = -36$. Now, we add these parts together: $a^2 + 2ab + b^2 = 36 + (-72i) + (-36)$. Combining the real parts ($36$ and $-36$) gives $36 - 36 = 0$. Combining the imaginary part ($-72i$) leaves us with $-72i$. Thus, the final simplified expression is $0 - 72i$, which simplifies to just $-72i$. This method, using the binomial expansion, provides a more structured way to approach the problem, especially for those familiar with algebraic identities. It confirms the result obtained through direct multiplication and reinforces the importance of correctly handling the imaginary unit $i$ and its square, $i^2$. Understanding this process is fundamental for operations involving powers of complex numbers, which are prevalent in fields like electrical engineering (for AC circuit analysis), quantum mechanics, and signal processing.

Alternative Method: Polar Form for Complex Number Exponentiation

While the binomial expansion or direct multiplication is straightforward for squaring a complex number, for higher powers, using the polar form of complex numbers becomes significantly more efficient. Let's explore how to simplify $(-6+6i)^2$ using this method, even though it might be a bit more involved for just a power of two. The polar form of a complex number $z = a + bi$ is given by $z = r( ext{cos}( heta) + i ext{sin}( heta))$, where $r$ is the magnitude (or modulus) and $ heta$ is the argument (or angle). For our complex number $z = -6 + 6i$, the real part $a = -6$ and the imaginary part $b = 6$. The magnitude $r$ is calculated as $r = \sqrt{a^2 + b^2} = \sqrt{(-6)^2 + (6)^2} = \sqrt{36 + 36} = \sqrt{72}$. We can simplify $\sqrt{72}$ as $\sqrt{36 imes 2} = 6\sqrt{2}$. The argument $ heta$ is found using $\text{tan}(\theta) = \frac{b}{a}$. So, $\text{tan}(\theta) = \frac{6}{-6} = -1$. Since the real part is negative and the imaginary part is positive, the complex number lies in the second quadrant. The principal value for $\text{arctan}(-1)$ is $-\frac{\pi}{4}$ (or $-45^\circ$). However, in the second quadrant, the angle is $\theta = \pi - \frac{\pi}{4} = \frac{3\pi}{4}$ (or $135^\circ$).

Now, according to De Moivre's Theorem, if $z = r( ext{cos}( heta) + i ext{sin}( heta))$, then $z^n = r^n( ext{cos}(n\theta) + i ext{sin}(n\theta))$. In our case, $n=2$. So, $(-6+6i)^2$ will be:

  • $r^2 = (6\sqrt{2})^2 = 36 imes 2 = 72$.
  • $2\theta = 2 imes \frac{3\pi}{4} = \frac{3\pi}{2}$.

Therefore, $(-6+6i)^2 = 72( ext{cos}(\frac{3\pi}{2}) + i ext{sin}(\frac{3\pi}{2}))$. We know that $\text{cos}(\frac{3\pi}{2}) = 0$ and $\text{sin}(\frac{3\pi}{2}) = -1$. Substituting these values, we get:

$(-6+6i)^2 = 72(0 + i(-1)) = 72(-i) = -72i$.

This confirms our previous results. While this method requires understanding polar coordinates and De Moivre's Theorem, it's incredibly powerful for simplifying complex numbers raised to higher integer powers, as it avoids tedious repeated multiplications. For example, calculating $(-6+6i)^{10}$ would be much simpler using the polar form.

Significance and Applications of Complex Number Operations

The ability to perform operations on complex numbers, such as simplifying expressions like $(-6+6i)^2$, is not just an academic exercise; it forms the backbone of many critical applications in science, technology, and engineering. Understanding how to manipulate complex numbers, especially their powers and roots, is essential for fields that deal with oscillations, waves, and alternating currents. For instance, in electrical engineering, complex numbers are used to represent AC voltages and currents, and their impedances. The impedance of a circuit component is a complex quantity that describes how it opposes the flow of alternating current. Simplifying expressions involving powers of complex numbers can help in analyzing the behavior of complex circuits, determining power consumption, and designing filters. In signal processing, complex numbers are fundamental to the Fourier Transform, a technique used to decompose signals into their constituent frequencies. The output of the Fourier Transform is a complex-valued function, and operations on these functions, including exponentiation, are routine. This allows engineers to analyze and manipulate signals in the frequency domain, leading to advancements in telecommunications, audio processing, and medical imaging.

Furthermore, in quantum mechanics, the state of a quantum system is described by a wave function, which is inherently complex-valued. The mathematics of quantum mechanics heavily relies on operations with complex numbers, including their powers, to predict the behavior of subatomic particles. The Schrödinger equation, a fundamental equation in quantum mechanics, involves complex numbers. In fluid dynamics, complex numbers can be used to model certain types of fluid flow, particularly in two-dimensional potential flow problems. The theory of functions of a complex variable provides powerful tools for analyzing such flows. Even in areas like fractal geometry, the generation of intricate and self-similar patterns, such as the Mandelbrot set, relies on repeatedly applying simple complex number operations, including squaring, to an initial set of points. The beauty and complexity of these fractals emerge from the iterative application of these basic mathematical rules. Therefore, mastering the simplification of complex number expressions is a key step towards understanding and contributing to these diverse and impactful fields.

For further exploration into the fascinating world of complex numbers and their applications, you might find the resources at Lumen Learning or the extensive mathematical explanations on LibreTexts to be incredibly insightful.