Simplify 10i(-7 + 17i) Express In A + Bi Form

In the realm of complex numbers, simplification is a fundamental skill. This article delves into the process of simplifying a complex expression and expressing it in the standard $a + bi$ form. Complex numbers, which extend the real number system by including the imaginary unit i (where $i^2 = -1$), play a crucial role in various fields of mathematics, physics, and engineering. Understanding how to manipulate and simplify these numbers is essential for solving complex equations and analyzing intricate systems. This guide provides a step-by-step approach to simplifying the expression $10i(-7 + 17i)$, ensuring clarity and comprehension for readers of all backgrounds. We will explore the distributive property, the definition of the imaginary unit, and the combination of like terms to arrive at the final simplified form. By mastering these techniques, you'll be well-equipped to tackle more advanced complex number problems and appreciate the elegance and utility of this mathematical concept. Whether you're a student learning the basics or a professional seeking a refresher, this article aims to provide a comprehensive understanding of complex number simplification.

Before diving into the simplification process, it's crucial to grasp the basics of complex numbers. A complex number is expressed in the form $a + bi$, where a represents the real part and b represents the imaginary part. The imaginary unit, denoted by i, is defined as the square root of -1, i.e., $i = \sqrt{-1}$. This implies that $i^2 = -1$, a fundamental property that we will use extensively in simplifying our expression. Complex numbers extend the real number system by incorporating this imaginary unit, allowing us to represent solutions to equations that have no real roots. For instance, the equation $x^2 + 1 = 0$ has no real solutions, but it has two complex solutions: $x = i$ and $x = -i$. The combination of a real part and an imaginary part allows complex numbers to represent points on a two-dimensional plane, known as the complex plane, where the horizontal axis represents the real part and the vertical axis represents the imaginary part. This geometric interpretation provides a powerful visual tool for understanding the behavior of complex numbers and their operations. Complex numbers are not just abstract mathematical constructs; they have practical applications in various fields. In electrical engineering, they are used to analyze alternating current (AC) circuits. In quantum mechanics, they are essential for describing wave functions. In fluid dynamics, they help in analyzing fluid flow. The ability to manipulate and simplify complex numbers is therefore a valuable skill in many scientific and engineering disciplines. Understanding the structure of complex numbers and the properties of the imaginary unit is the first step towards mastering complex number arithmetic. We will build upon this foundation as we proceed with the simplification of the given expression.

Our main task is to simplify the complex expression $10i(-7 + 17i)$ and express the result in the standard $a + bi$ form. This involves distributing the term $10i$ across the parentheses and then combining like terms. The $a + bi$ form is the standard way to represent complex numbers, where a is the real part and b is the imaginary part. To achieve this, we'll need to apply the distributive property of multiplication over addition, which states that $a(b + c) = ab + ac$. In our case, $10i$ will be multiplied by both $-7$ and $17i$. This process will generate terms involving i and $i^2$. Remember that $i^2$ is equal to -1, which is a key identity for simplifying complex expressions. After distributing and simplifying, we will group the real and imaginary terms together. The real terms will form the a part of our $a + bi$ form, and the imaginary terms will form the bi part. This step is crucial for presenting the final answer in the required format. The challenge lies in correctly applying the distributive property and simplifying the terms involving i. Mistakes in these steps can lead to an incorrect final answer. Therefore, we will proceed carefully and methodically, explaining each step in detail. Understanding the problem statement clearly sets the stage for the simplification process. We know what we need to achieve (the $a + bi$ form) and the tools we need to use (distributive property, $i^2 = -1$). With a clear understanding of the goal, we can move on to the step-by-step simplification process, ensuring that each step is logically sound and mathematically accurate. This structured approach will help in minimizing errors and maximizing comprehension.

To begin, we apply the distributive property to the expression $10i(-7 + 17i)$. This means we multiply $10i$ by both terms inside the parentheses:

$10i(-7) + 10i(17i)$

This simplifies to:

$-70i + 170i^2$

Next, we use the fundamental property of the imaginary unit: $i^2 = -1$. Substituting this into our expression, we get:

$-70i + 170(-1)$

This further simplifies to:

$-70i - 170$

Now, we rearrange the terms to express the result in the standard $a + bi$ form, where a is the real part and b is the imaginary part:

$-170 - 70i$

Therefore, the simplified form of $10i(-7 + 17i)$ is $-170 - 70i$. Each step in this process is crucial. The distributive property allows us to break down the complex expression into simpler terms. The substitution of $i^2$ with -1 is the key to eliminating the imaginary unit from the squared term and converting it into a real number. Finally, rearranging the terms into the $a + bi$ form ensures that our answer is in the standard format for complex numbers. This step-by-step approach not only leads to the correct solution but also provides a clear understanding of the underlying principles and techniques involved in simplifying complex expressions. By carefully following each step and understanding the reasoning behind it, you can confidently tackle similar problems and develop a strong foundation in complex number arithmetic. The ability to simplify complex expressions is a fundamental skill in various mathematical and scientific contexts, making this exercise a valuable learning experience.

After following the step-by-step simplification process, we have arrived at the final answer in $a + bi$ form: $-170 - 70i$. This result clearly shows the real part as -170 and the imaginary part as -70. The $a + bi$ form is crucial for representing complex numbers in a standardized way, making it easier to perform further operations and comparisons. The real part (-170) lies along the real axis in the complex plane, while the imaginary part (-70) lies along the imaginary axis. This geometric interpretation provides a visual representation of the complex number's location and magnitude. The negative signs indicate that both the real and imaginary components are negative, placing the number in the third quadrant of the complex plane. The process of arriving at this final answer involved several key steps: applying the distributive property, substituting $i^2$ with -1, and rearranging the terms into the $a + bi$ form. Each of these steps is essential for accurate simplification. Omitting or misapplying any step can lead to an incorrect result. The final answer, $-170 - 70i$, is not only mathematically correct but also provides a clear and concise representation of the simplified complex expression. This form allows for easy identification of the real and imaginary components, which is crucial for various applications. Whether you're solving equations, analyzing circuits, or exploring quantum mechanics, the ability to express complex numbers in the standard $a + bi$ form is a fundamental skill. Therefore, mastering this simplification process is a valuable asset in your mathematical toolkit. This final answer serves as a culmination of the simplification process and a testament to the power of complex number arithmetic.

In this comprehensive guide, we have successfully simplified the complex expression $10i(-7 + 17i)$ and expressed it in the standard $a + bi$ form, which is $-170 - 70i$. We began by understanding the fundamentals of complex numbers, including the definition of the imaginary unit i and the significance of the $a + bi$ form. We then meticulously applied the distributive property, substituted $i^2$ with -1, and combined like terms to arrive at the final simplified expression. This step-by-step approach not only provided the correct answer but also elucidated the underlying principles and techniques involved in simplifying complex numbers. The ability to simplify complex expressions is a crucial skill in various fields, including mathematics, physics, and engineering. Complex numbers are used to model a wide range of phenomena, from alternating current circuits to quantum mechanical wave functions. Therefore, mastering complex number arithmetic is essential for anyone working in these disciplines. The process we followed in this article can be applied to a variety of complex expressions, making it a valuable tool for problem-solving. By understanding the distributive property, the properties of the imaginary unit, and the importance of the $a + bi$ form, you can confidently tackle more complex problems and deepen your understanding of complex numbers. This guide has aimed to provide a clear and accessible explanation of the simplification process, ensuring that readers of all backgrounds can grasp the concepts and apply them effectively. The final result, $-170 - 70i$, stands as a testament to the power and elegance of complex number arithmetic. We hope that this article has empowered you with the knowledge and skills to simplify complex expressions and appreciate the beauty of complex numbers.