Equivalent Expressions Of Complex Numbers A Detailed Explanation

Introduction

In the realm of mathematics, expressions often appear in various forms, yet some may possess an underlying equivalence. This article delves into the intricacies of complex number expressions, specifically focusing on identifying equivalent forms of a given expression. We will dissect the expression $(-\sqrt{9}+\sqrt{-4})-(2 \sqrt{576}+\sqrt{-64})$, unraveling its components, and meticulously comparing it with a set of potential equivalents. By adhering to the fundamental principles of complex number arithmetic, we aim to illuminate the path toward accurate evaluation and equivalence determination. Understanding these concepts is crucial for anyone venturing into advanced mathematical studies, particularly in fields like electrical engineering, quantum mechanics, and signal processing, where complex numbers play a pivotal role. This exploration will not only enhance your ability to manipulate complex expressions but also deepen your comprehension of the mathematical structures that govern these fascinating numbers. By carefully examining each component and applying the rules of arithmetic, we will navigate through the complexities of this expression and reveal its true form.

Dissecting the Given Expression

To embark on our quest for equivalent expressions, we must first dissect the given expression: $(-\sqrt{9}+\sqrt{-4})-(2 \sqrt{576}+\sqrt{-64})$. This expression intricately weaves together real and imaginary components, demanding a keen understanding of square roots, negative numbers, and complex number arithmetic. Our initial step involves simplifying each term individually, paying meticulous attention to the order of operations and the properties of square roots. The term $-\sqrt{9}$ is straightforward; the square root of 9 is 3, thus this term simplifies to -3. Next, we encounter $\sqrt{-4}$, which introduces us to the realm of imaginary numbers. Recall that the square root of -1 is denoted as i, the imaginary unit. Therefore, $\sqrt{-4}$ can be expressed as $\sqrt{4 \cdot -1}$, which further simplifies to $\sqrt{4} \cdot \sqrt{-1}$, or 2i. Shifting our focus to the second part of the expression, we have $2 \sqrt{576}$. We must determine the square root of 576, which is 24. Multiplying this by 2, we get 2 * 24, or 48. Finally, we encounter $\sqrt{-64}$, another imaginary term. Similar to our approach with $\sqrt{-4}$, we express $\sqrt{-64}$ as $\sqrt{64 \cdot -1}$, which simplifies to $\sqrt{64} \cdot \sqrt{-1}$, or 8i. With each term simplified, we can now rewrite the expression as (-3 + 2i) - (48 + 8i). This form allows us to clearly see the real and imaginary parts, setting the stage for the next step: combining like terms.

Simplifying the Expression

The next crucial step in our exploration is to simplify the expression we've dissected. Having broken down the complex expression $(-\sqrt{9}+\sqrt{-4})-(2 \sqrt{576}+\sqrt{-64})$ into its constituent parts, we arrived at (-3 + 2i) - (48 + 8i). Now, we focus on removing the parentheses and combining like terms, a fundamental process in algebraic simplification. The first set of parentheses, (-3 + 2i), can be simply removed without altering the terms inside. However, the second set of parentheses, (48 + 8i), is preceded by a subtraction sign, which necessitates distributing the negative sign to each term within. This means we subtract both 48 and 8i. Consequently, the expression transforms into -3 + 2i - 48 - 8i. With the parentheses removed, we can now identify and combine the real and imaginary terms separately. The real terms are -3 and -48, which combine to give -51. The imaginary terms are 2i and -8i, which combine to give -6i. Therefore, the simplified expression is -51 - 6i. This resulting complex number is now in its standard form, where the real part is -51 and the imaginary part is -6. This simplification process is not just a mathematical exercise; it is a crucial step in making the expression more understandable and comparable to other complex numbers. By reducing the expression to its simplest form, we eliminate any ambiguity and lay the groundwork for accurately identifying equivalent expressions. The simplified form, -51 - 6i, serves as our benchmark against which we will evaluate the given options, ensuring we select only those that truly represent the same complex number.

Evaluating Potential Equivalents

Having simplified the original expression to -51 - 6i, we now turn our attention to evaluating the potential equivalents provided. This stage is crucial as it involves comparing each option with our simplified form to determine which ones truly match. We must meticulously examine each option, applying the rules of arithmetic and complex number manipulation to bring them into a form comparable to -51 - 6i. The first option is -3 + 2i + 2(24) + 8i. To evaluate this, we first perform the multiplication: 2(24) equals 48. The expression then becomes -3 + 2i + 48 + 8i. Combining the real terms (-3 and 48) yields 45, and combining the imaginary terms (2i and 8i) yields 10i. Thus, this option simplifies to 45 + 10i, which is clearly not equivalent to -51 - 6i. The second option is -51 + 6i. This expression has the same real part as our simplified form (-51), but the imaginary part is +6i, while our simplified expression has -6i. Therefore, this option is not equivalent. The third option is -3 - 2i - 2(24) + 8i. Again, we start by performing the multiplication: 2(24) equals 48. The expression becomes -3 - 2i - 48 + 8i. Combining the real terms (-3 and -48) gives -51, and combining the imaginary terms (-2i and 8i) gives 6i. So, this option simplifies to -51 + 6i, which is also not equivalent to -51 - 6i. The fourth option is -3 + 2i - 2(24) - 8i. Multiplying 2 by 24 gives 48, so the expression becomes -3 + 2i - 48 - 8i. Combining the real terms (-3 and -48) results in -51, and combining the imaginary terms (2i and -8i) gives -6i. This option simplifies to -51 - 6i, which exactly matches our simplified form. Therefore, this option is equivalent. The fifth option is -51 - 6i. This option is identical to our simplified form, making it a clear equivalent. Finally, the sixth option is 45 + 10i, which we previously identified as not equivalent. By methodically evaluating each option, we have successfully identified the ones that are equivalent to the original expression.

Identifying Equivalent Expressions

Through our meticulous evaluation process, we have now reached the critical juncture of identifying the expressions that are indeed equivalent to our simplified form, -51 - 6i. This process is not merely about finding matching terms; it's about affirming that the entire complex number, both its real and imaginary components, aligns perfectly. Our journey began with the complex expression $(-\sqrt{9}+\sqrt{-4})-(2 \sqrt{576}+\sqrt{-64})$, which we painstakingly simplified to -51 - 6i. This simplified form became our benchmark, the gold standard against which we measured each potential equivalent. In our evaluation, we discovered that the fourth option, -3 + 2i - 2(24) - 8i, after undergoing simplification, yielded -51 - 6i. This perfect match confirms its equivalence to the original expression. Furthermore, the fifth option, -51 - 6i, presented itself as an exact replica of our simplified form, leaving no room for doubt about its equivalence. These two options, and only these two, have demonstrated their fidelity to the original expression by converging to the same simplified complex number. The other options, despite their superficial similarities, diverged in either their real or imaginary components, disqualifying them from being considered equivalent. This careful selection process underscores the importance of precision in complex number arithmetic and highlights the need to consider both the magnitude and sign of each component. By identifying these equivalent expressions, we not only solve the given problem but also reinforce our understanding of complex number equivalence, a cornerstone concept in advanced mathematics and its applications.

Conclusion

In conclusion, our exploration into the realm of complex number expressions has culminated in the successful identification of equivalent forms for the expression $(-\sqrt{9}+\sqrt{-4})-(2 \sqrt{576}+\sqrt{-64})$. We embarked on this journey by meticulously dissecting the given expression, simplifying each component to its most basic form. This involved navigating the intricacies of square roots, imaginary units, and the order of operations. Through careful application of arithmetic principles, we reduced the original expression to its simplified form: -51 - 6i. This simplified form served as our definitive benchmark, the yardstick against which we measured the potential equivalents. We then rigorously evaluated each of the provided options, subjecting them to the same simplification process. This involved distributing negative signs, combining like terms, and paying close attention to the signs and magnitudes of both real and imaginary components. Our diligent evaluation revealed that only two options, -3 + 2i - 2(24) - 8i and -51 - 6i, perfectly matched our simplified form. These expressions, and only these, were deemed equivalent to the original. This exercise underscores the critical importance of precision and attention to detail in complex number arithmetic. It highlights the fact that expressions can appear different on the surface yet possess an underlying equivalence. Identifying these equivalencies is not just a mathematical skill; it's a fundamental concept that permeates various fields, from electrical engineering to quantum physics. By mastering these techniques, we not only enhance our mathematical proficiency but also equip ourselves with the tools to tackle complex problems in a wide array of disciplines. The journey through this problem has reinforced the power of simplification, the necessity of careful evaluation, and the profound connections between seemingly disparate mathematical expressions.