Simplify The Expression (3i + 4) - I + 4(6 + 2i) A Comprehensive Guide

In the realm of mathematics, particularly when dealing with complex numbers, the ability to simplify expressions is paramount. This article delves into the process of simplifying the expression $(3i + 4) - i + 4(6 + 2i)$, providing a step-by-step guide that not only arrives at the correct answer but also illuminates the underlying principles of complex number arithmetic. Understanding these principles is crucial for anyone venturing into fields like electrical engineering, quantum mechanics, and advanced mathematics, where complex numbers are ubiquitous.

The Essence of Complex Numbers

Before we embark on the simplification journey, let's briefly revisit the concept of complex numbers. A complex number is composed of two parts: a real part and an imaginary part. It is typically represented in the form $a + bi$, where '$a$' denotes the real part, '$b$' represents the imaginary part, and '$i$' is the imaginary unit, defined as the square root of -1 ($\sqrt{-1}$). The imaginary unit '$i$' is the cornerstone of complex number arithmetic, allowing us to perform operations that are not possible with real numbers alone. Complex numbers find applications in diverse fields, including electrical engineering for circuit analysis, quantum mechanics for describing wave functions, and signal processing for analyzing signals.

Deconstructing the Expression

The expression we aim to simplify is $(3i + 4) - i + 4(6 + 2i)$. At first glance, it might seem daunting, but by breaking it down into smaller, manageable steps, the solution becomes clear. The expression involves addition, subtraction, and multiplication of complex numbers, all of which follow specific rules. Our approach will be methodical, ensuring that each step is executed with precision.

Step 1: Distribute the Scalar

The first step in simplifying the expression is to distribute the scalar '4' across the terms within the parentheses in the expression $4(6 + 2i)$. Scalar multiplication in complex numbers is analogous to scalar multiplication in real numbers. We multiply the scalar by both the real and imaginary parts of the complex number. This gives us:

$4 * 6 = 24$ $4 * 2i = 8i$

Thus, $4(6 + 2i)$ simplifies to $24 + 8i$. This step is crucial as it eliminates the parentheses, making the expression easier to manage. The distributive property is a fundamental concept in algebra and is equally applicable in the context of complex numbers. Mastering this step is essential for simplifying more complex expressions involving complex numbers.

Step 2: Rewrite the Expression

Now that we have distributed the scalar, we can rewrite the original expression as:

$(3i + 4) - i + 24 + 8i$

This rewritten expression is more streamlined and sets the stage for the next step, which involves combining like terms. By eliminating the parentheses and performing the scalar multiplication, we have transformed the expression into a form that is easier to manipulate. This step highlights the importance of order of operations in mathematics, ensuring that we perform operations in the correct sequence to arrive at the correct answer.

Step 3: Combine Like Terms

The next step is to combine the 'like terms'. In the context of complex numbers, 'like terms' refer to the real parts and the imaginary parts. We group the real numbers together and the imaginary numbers together. This involves identifying the terms that contain the imaginary unit '$i$' and those that do not.

Grouping the Real Parts:

We have the real numbers 4 and 24. Combining these gives:

$4 + 24 = 28$

Grouping the Imaginary Parts:

We have the imaginary terms $3i$, $-i$, and $8i$. Combining these gives:

$3i - i + 8i = (3 - 1 + 8)i = 10i$

Combining like terms is a fundamental algebraic technique that simplifies expressions by reducing the number of terms. In the context of complex numbers, it allows us to isolate the real and imaginary components, making the expression more concise and easier to interpret.

Step 4: Write in Standard Form

Finally, we write the simplified expression in the standard form of a complex number, which is $a + bi$, where '$a$' is the real part and '$b$' is the imaginary part. From the previous step, we have the real part as 28 and the imaginary part as $10i$. Therefore, the simplified expression is:

$28 + 10i$

This is the final simplified form of the original expression. Writing complex numbers in standard form is a convention that ensures clarity and consistency in mathematical communication. It allows for easy comparison and manipulation of complex numbers in various applications.

In conclusion, the simplification of the expression $(3i + 4) - i + 4(6 + 2i)$ demonstrates the fundamental principles of complex number arithmetic. By meticulously following the steps of distributing scalars, combining like terms, and expressing the result in standard form, we arrive at the simplified expression $28 + 10i$. This process underscores the importance of understanding the properties of complex numbers and the rules that govern their manipulation.

The ability to simplify complex expressions is not merely an academic exercise; it is a crucial skill in various fields that rely on complex number theory. Whether you are an engineer designing circuits, a physicist exploring quantum phenomena, or a mathematician delving into abstract algebra, a solid grasp of complex number arithmetic is indispensable. By mastering the techniques outlined in this article, you will be well-equipped to tackle more complex problems and appreciate the elegance and power of complex numbers.

This comprehensive guide has hopefully demystified the process of simplifying complex expressions. Remember, practice is key to mastery. The more you work with complex numbers, the more intuitive these operations will become. So, keep exploring, keep practicing, and keep simplifying!

To effectively address the task of simplifying the expression $(3i + 4) - i + 4(6 + 2i)$, a systematic approach is crucial. This involves not only understanding the individual components of the expression but also recognizing the order of operations and the properties of complex numbers. Let's break down the problem into manageable segments to ensure a clear and accurate solution. This meticulous approach is vital in mathematics, especially when dealing with complex numbers where precision is paramount.

Identifying the Components:

The expression consists of several terms involving complex numbers and real numbers. We have the complex number $(3i + 4)$, the imaginary unit $-i$, and the product of a scalar and a complex number $4(6 + 2i)$. Each of these components plays a distinct role in the expression, and understanding their individual nature is the first step towards simplification. Complex numbers, as we discussed earlier, have a real part and an imaginary part, and operations involving them require careful attention to both parts.

Order of Operations:

The order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), dictates the sequence in which mathematical operations should be performed. In our expression, we first need to address the multiplication within the parentheses before proceeding with addition and subtraction. This ensures that we maintain mathematical consistency and arrive at the correct answer. Ignoring the order of operations can lead to significant errors, especially in complex expressions.

Properties of Complex Numbers:

Complex numbers have unique properties that govern their behavior under various operations. For instance, the imaginary unit '$i$' has the property that $i^2 = -1$. While this property is not directly relevant in this specific problem, it is crucial in more complex scenarios involving powers of '$i$'. Understanding the properties of complex numbers is essential for manipulating them effectively and avoiding common pitfalls.

Recognizing the Goal:

The ultimate goal is to simplify the expression, which means combining like terms and expressing the result in the standard form of a complex number, $a + bi$. This involves identifying the real and imaginary parts of the expression and consolidating them into a single complex number. Simplification not only makes the expression more concise but also facilitates further calculations and analysis.

A Strategic Approach:

With a clear understanding of the components, order of operations, and properties of complex numbers, we can devise a strategic approach to simplify the expression. This involves the following steps:

1. Distribute the scalar: Multiply the scalar '4' by the complex number $(6 + 2i)$.
2. Rewrite the expression: Substitute the result of the scalar multiplication back into the original expression.
3. Combine like terms: Group the real parts and the imaginary parts separately.
4. Express in standard form: Write the simplified expression in the form $a + bi$.

By following this methodical approach, we can systematically simplify the expression and arrive at the correct answer. Each step builds upon the previous one, ensuring that we maintain accuracy and clarity throughout the process. This strategic mindset is invaluable in mathematics and problem-solving in general.

Anticipating Challenges:

While the simplification process is straightforward, it is essential to anticipate potential challenges. One common mistake is incorrectly distributing the scalar or combining unlike terms. Another challenge is maintaining the correct signs when performing addition and subtraction. By being mindful of these potential pitfalls, we can minimize errors and ensure a successful outcome.

In summary, understanding the problem involves dissecting the expression into its components, recognizing the order of operations, understanding the properties of complex numbers, and devising a strategic approach. By meticulously addressing each of these aspects, we can confidently embark on the simplification journey and arrive at the correct solution.

Having dissected the problem and laid out a strategic approach, we now embark on the step-by-step solution to simplify the expression $(3i + 4) - i + 4(6 + 2i)$. This detailed walkthrough will not only provide the solution but also elucidate the reasoning behind each step, reinforcing your understanding of complex number arithmetic. Each step is crucial in the simplification process, and a thorough understanding of each will enhance your mathematical proficiency.

Step 1: Distribute the Scalar

The first step in simplifying the expression is to distribute the scalar '4' across the terms within the parentheses in the expression $4(6 + 2i)$. This involves multiplying '4' by both the real part (6) and the imaginary part (2i). This is a fundamental application of the distributive property, which states that $a(b + c) = ab + ac$. In the context of complex numbers, this property holds true, allowing us to distribute scalars across complex terms.

$4 * 6 = 24$ $4 * 2i = 8i$

Therefore, $4(6 + 2i)$ simplifies to $24 + 8i$. This step is crucial as it eliminates the parentheses, making the expression easier to manage. The distributive property is a cornerstone of algebra, and its application here demonstrates its versatility in handling complex numbers.

Step 2: Rewrite the Expression

Now that we have distributed the scalar, we can rewrite the original expression as:

$(3i + 4) - i + 24 + 8i$

This rewritten expression is more streamlined and sets the stage for the next step, which involves combining like terms. By eliminating the parentheses and performing the scalar multiplication, we have transformed the expression into a form that is easier to manipulate. This step highlights the importance of order of operations in mathematics, ensuring that we perform operations in the correct sequence to arrive at the correct answer. The careful rewriting of the expression is a testament to the importance of clarity in mathematical problem-solving.

Step 3: Combine Like Terms

The next step is to combine the 'like terms'. In the context of complex numbers, 'like terms' refer to the real parts and the imaginary parts. We group the real numbers together and the imaginary numbers together. This involves identifying the terms that contain the imaginary unit '$i$' and those that do not. This process of grouping like terms is a fundamental algebraic technique that simplifies expressions by reducing the number of terms.

Grouping the Real Parts:

We have the real numbers 4 and 24. Combining these gives:

$4 + 24 = 28$

Grouping the Imaginary Parts:

We have the imaginary terms $3i$, $-i$, and $8i$. Combining these gives:

$3i - i + 8i = (3 - 1 + 8)i = 10i$

Combining like terms is a fundamental algebraic technique that simplifies expressions by reducing the number of terms. In the context of complex numbers, it allows us to isolate the real and imaginary components, making the expression more concise and easier to interpret. This step demonstrates the power of algebraic manipulation in simplifying complex expressions.

Step 4: Write in Standard Form

Finally, we write the simplified expression in the standard form of a complex number, which is $a + bi$, where '$a$' is the real part and '$b$' is the imaginary part. From the previous step, we have the real part as 28 and the imaginary part as $10i$. Therefore, the simplified expression is:

$28 + 10i$

This is the final simplified form of the original expression. Writing complex numbers in standard form is a convention that ensures clarity and consistency in mathematical communication. It allows for easy comparison and manipulation of complex numbers in various applications. This final step underscores the importance of adhering to mathematical conventions for clear and effective communication.

By meticulously following these steps, we have successfully simplified the expression $(3i + 4) - i + 4(6 + 2i)$ to $28 + 10i$. This detailed walkthrough not only provides the solution but also highlights the underlying principles and techniques involved in complex number arithmetic. Mastering these steps will equip you to tackle more complex problems and appreciate the elegance and power of complex numbers.

Now that we have meticulously simplified the expression $(3i + 4) - i + 4(6 + 2i)$ to $28 + 10i$, it is crucial to verify our solution against the given options. This step ensures that we have not only arrived at the correct answer but also understood the process thoroughly. Analyzing the options also provides insights into common errors and misconceptions, further solidifying our understanding of the topic.

The Simplified Expression:

As we have demonstrated in the step-by-step solution, the simplified form of the expression is:

$28 + 10i$

This complex number has a real part of 28 and an imaginary part of 10i. It is essential to remember that complex numbers are expressed in the standard form $a + bi$, where 'a' is the real part and 'bi' is the imaginary part. This standard form facilitates comparison and manipulation of complex numbers.

Analyzing the Options:

Let's examine the given options:

A. $14 + 4i$ B. $28 + 10i$ C. $27 + 11i$ D. $14 + 8i$

By comparing our simplified expression with the options, we can clearly identify the correct answer.

Identifying the Correct Option:

The correct option is:

B. $28 + 10i$

This option perfectly matches our simplified expression, confirming the accuracy of our solution. The real part (28) and the imaginary part (10i) are both consistent with our result.

Analyzing Incorrect Options:

To further enhance our understanding, let's analyze why the other options are incorrect. This will help us identify potential errors and misconceptions that might arise during the simplification process.

A. $14 + 4i$: This option is incorrect because it significantly deviates from our simplified expression. The real part (14) and the imaginary part (4i) do not match our calculated values. This could be due to errors in distributing the scalar or combining like terms.

C. $27 + 11i$: This option is close to the correct answer but has slight variations in both the real and imaginary parts. The real part (27) is one less than the correct value (28), and the imaginary part (11i) is one more than the correct value (10i). This could be due to minor arithmetic errors during the simplification process.

D. $14 + 8i$: This option, like option A, is incorrect due to significant deviations from our simplified expression. The real part (14) is incorrect, and the imaginary part (8i) is also different from our calculated value (10i). This could be due to a combination of errors in scalar distribution, combining like terms, or other arithmetic mistakes.

Key Takeaways:

This analysis highlights the importance of meticulous step-by-step simplification and careful comparison with the given options. Common errors include:

• Incorrectly distributing scalars.
• Combining unlike terms.
• Making arithmetic mistakes during addition and subtraction.

By being aware of these potential pitfalls, we can minimize errors and ensure accurate solutions.

In conclusion, by comparing our simplified expression with the given options, we have confidently identified the correct answer as $28 + 10i$. The analysis of incorrect options has provided valuable insights into common errors and misconceptions, further reinforcing our understanding of complex number arithmetic. This comprehensive approach ensures not only the correct solution but also a deeper appreciation of the mathematical principles involved.

The simplification of complex expressions, as demonstrated in the problem $(3i + 4) - i + 4(6 + 2i)$, might seem like an abstract mathematical exercise. However, complex numbers and their simplification have profound real-world applications across various scientific and engineering disciplines. Understanding these applications underscores the importance of mastering complex number arithmetic and its role in solving practical problems.

Electrical Engineering:

One of the most prominent applications of complex numbers is in electrical engineering, particularly in the analysis of alternating current (AC) circuits. In AC circuits, voltage and current vary sinusoidally with time, and complex numbers provide a convenient way to represent these sinusoidal quantities as phasors. Phasors are complex numbers that represent the amplitude and phase of a sinusoidal function. The impedance of a circuit, which is the opposition to the flow of current, is also a complex quantity.

Simplifying complex expressions is crucial in circuit analysis for determining the total impedance of a circuit, calculating current and voltage values, and analyzing power consumption. For example, when dealing with series and parallel combinations of resistors, capacitors, and inductors, the total impedance is calculated using complex number arithmetic. The ability to simplify these expressions is essential for designing and analyzing electrical circuits.

Quantum Mechanics:

Complex numbers play a fundamental role in quantum mechanics, the theory that governs the behavior of matter at the atomic and subatomic levels. In quantum mechanics, the state of a system is described by a wave function, which is a complex-valued function. The wave function contains all the information about the system, such as its energy, momentum, and position.

The Schrödinger equation, the fundamental equation of quantum mechanics, involves complex numbers and complex-valued operators. Solving the Schrödinger equation often requires simplifying complex expressions. For example, when calculating the probability of finding a particle in a particular state, complex number arithmetic is essential. The simplification of complex expressions is thus crucial for understanding and predicting the behavior of quantum systems.

Signal Processing:

Complex numbers are widely used in signal processing, a field that deals with the analysis and manipulation of signals, such as audio, images, and communication signals. The Fourier transform, a fundamental tool in signal processing, transforms a signal from the time domain to the frequency domain, representing the signal as a sum of complex exponentials.

Simplifying complex expressions is crucial in signal processing for analyzing the frequency content of signals, designing filters, and processing digital images. For example, when designing a filter to remove noise from an audio signal, complex number arithmetic is used to determine the filter's characteristics. The ability to simplify complex expressions is essential for developing efficient and effective signal processing algorithms.

Control Systems:

Complex numbers are also used in control systems, which are systems that regulate the behavior of other systems. Control systems are used in a wide range of applications, such as robotics, aerospace, and industrial automation. The stability and performance of a control system are often analyzed using complex numbers and techniques such as the Nyquist stability criterion.

Simplifying complex expressions is crucial in control systems for designing controllers that ensure stability and meet performance requirements. For example, when designing a controller for a robot arm, complex number arithmetic is used to analyze the system's response to disturbances and ensure that the arm moves smoothly and accurately. The ability to simplify complex expressions is essential for developing robust and reliable control systems.

Other Applications:

Besides the applications mentioned above, complex numbers and their simplification are used in various other fields, including:

• Fluid dynamics: For analyzing fluid flow and wave propagation.
• Electromagnetism: For describing electromagnetic waves and fields.
• Mathematics: In complex analysis, a branch of mathematics that deals with complex-valued functions.

In conclusion, the simplification of complex expressions is not merely an abstract mathematical exercise but a crucial skill with wide-ranging real-world applications. From electrical engineering to quantum mechanics, signal processing to control systems, complex numbers and their simplification play a vital role in solving practical problems and advancing scientific and technological progress. Mastering complex number arithmetic is thus an invaluable asset for anyone pursuing a career in these fields.

In this comprehensive exploration, we have delved into the intricate process of simplifying the complex expression $(3i + 4) - i + 4(6 + 2i)$. We embarked on this journey by first understanding the fundamental principles of complex numbers, their components, and their arithmetic operations. We then meticulously dissected the problem, recognizing the order of operations and the properties that govern complex number manipulation. This thorough preparation laid the groundwork for a systematic and accurate solution.

Our step-by-step solution meticulously walked through the process of distributing scalars, combining like terms, and expressing the result in the standard form of a complex number, $a + bi$. Each step was carefully explained, elucidating the reasoning behind the mathematical operations and reinforcing the underlying concepts. This detailed walkthrough served not only to provide the correct answer but also to instill a deeper understanding of the techniques involved in simplifying complex expressions.

We further validated our solution by comparing it against the given options, identifying the correct answer and analyzing the incorrect ones. This critical analysis provided valuable insights into common errors and misconceptions, further solidifying our grasp of the topic. By understanding why certain options were incorrect, we gained a more nuanced appreciation of the nuances of complex number arithmetic.

Beyond the immediate problem, we explored the profound real-world applications of complex numbers and their simplification across various scientific and engineering disciplines. From electrical engineering and quantum mechanics to signal processing and control systems, complex numbers play a vital role in solving practical problems and driving technological advancements. This exploration underscored the importance of mastering complex number arithmetic as an essential skill for anyone pursuing a career in these fields.

The art of simplifying complex expressions is not merely about memorizing rules and procedures; it is about developing a deep understanding of the underlying mathematical principles and applying them strategically. It requires a combination of precision, attention to detail, and a systematic approach. By mastering these skills, you will be well-equipped to tackle more complex problems and appreciate the elegance and power of complex numbers.

As we conclude this comprehensive guide, it is important to remember that practice is the key to mastery. The more you work with complex numbers, the more intuitive these operations will become. So, continue to explore, continue to practice, and continue to simplify. The world of complex numbers is vast and fascinating, and the ability to navigate it with confidence is a valuable asset in the modern world.

In summary, this exploration has provided a comprehensive understanding of simplifying complex expressions, from the foundational principles to the real-world applications. By mastering the techniques and concepts discussed, you will be well-prepared to tackle a wide range of mathematical challenges and appreciate the beauty and power of complex numbers.

B. $28 + 10i$