Equivalent Expression Of (4 + 7i)(3 + 4i) A Step-by-Step Guide

In the realm of mathematics, complex numbers extend the familiar number system by incorporating the imaginary unit, denoted as 'i', which is defined as the square root of -1. Complex numbers are expressed in the form a + bi, where a and b are real numbers, and i is the imaginary unit. Multiplying complex numbers involves a process similar to multiplying binomials, with the added consideration of simplifying terms involving i^2, which equals -1. This article delves into the multiplication of complex numbers, specifically focusing on the expression (4 + 7i)(3 + 4i), to determine the equivalent form from the given options. We will explore the step-by-step process of multiplying these complex numbers, simplifying the result, and identifying the correct answer choice. The ability to manipulate complex numbers is fundamental in various fields such as electrical engineering, quantum mechanics, and applied mathematics, making this a crucial concept to grasp. The multiplication of complex numbers is a fundamental operation in various fields of mathematics, physics, and engineering. It's essential for solving problems involving alternating current circuits, quantum mechanics, and signal processing. Understanding how to multiply complex numbers accurately is crucial for success in these areas. In this article, we will explore the multiplication of two complex numbers, (4 + 7i) and (3 + 4i), step-by-step, and identify the equivalent expression from the given options.

Step-by-Step Multiplication Process

To find the equivalent expression for (4 + 7i)(3 + 4i), we apply the distributive property, often remembered by the acronym FOIL (First, Outer, Inner, Last). This method ensures that each term in the first complex number is multiplied by each term in the second complex number. Let's break down the process:

1. First: Multiply the first terms in each complex number: 4 * 3 = 12
2. Outer: Multiply the outer terms: 4 * 4i = 16i
3. Inner: Multiply the inner terms: 7i * 3 = 21i
4. Last: Multiply the last terms: 7i * 4i = 28i^2

Combining these results, we get:

12 + 16i + 21i + 28i^2

Simplifying the Expression

The next step involves simplifying the expression. We know that i^2 = -1, so we can substitute -1 for i^2 in the expression:

12 + 16i + 21i + 28(-1)

This simplifies to:

12 + 16i + 21i - 28

Now, we combine the real parts (12 and -28) and the imaginary parts (16i and 21i):

Real part: 12 - 28 = -16

Imaginary part: 16i + 21i = 37i

Therefore, the simplified expression is:

-16 + 37i

This result matches option A, which is -16 + 37i. The process of multiplying complex numbers is similar to multiplying binomials, but it also involves simplifying the terms with i^2. The distributive property is applied to multiply each term of the first complex number by each term of the second complex number. After expanding the expression, we substitute -1 for i^2 and combine like terms. This simplification leads us to the equivalent complex number in the standard form a + bi. This standard form makes it easier to perform further operations or comparisons with other complex numbers. Understanding the properties of complex numbers and how they interact with algebraic operations is crucial for solving more complex mathematical problems and real-world applications. The accuracy in each step of the multiplication and simplification process is paramount to arriving at the correct answer. By meticulously applying the distributive property and correctly substituting i^2 with -1, we can confidently determine the equivalent complex number. This meticulous approach not only helps in solving mathematical problems but also reinforces the understanding of complex number arithmetic.

Common Mistakes to Avoid

When multiplying complex numbers, several common mistakes can lead to incorrect answers. Recognizing and avoiding these pitfalls is crucial for accuracy:

1. Forgetting to Distribute: One of the most common errors is failing to multiply each term in the first complex number by every term in the second complex number. The FOIL method helps prevent this by ensuring that all combinations are accounted for.
2. Incorrectly Simplifying i^2: Remember that i^2 = -1. A frequent mistake is either ignoring this substitution or miscalculating it. Always replace i^2 with -1 to simplify the expression correctly.
3. Combining Real and Imaginary Terms Prematurely: Ensure that you only combine real terms with real terms and imaginary terms with imaginary terms. Mixing these can lead to an incorrect result.
4. Sign Errors: Pay close attention to the signs, especially when dealing with negative numbers and the imaginary unit. A small sign error can significantly alter the final answer.
5. Overlooking the Standard Form: The final answer should be in the standard form a + bi. Make sure to separate the real and imaginary parts clearly.

By being mindful of these potential errors and carefully reviewing each step, you can increase your accuracy when multiplying complex numbers. To master the multiplication of complex numbers, it is beneficial to practice regularly and review the steps involved. Each problem solved reinforces the understanding of the process and helps in identifying and correcting any misunderstandings. The consistent application of the multiplication rules and simplification techniques builds confidence and accuracy in handling complex number operations. Furthermore, understanding the conceptual basis of complex numbers, including the definition of the imaginary unit and its properties, is essential for solving more complex problems and applications. This conceptual understanding allows for a more intuitive approach to problem-solving and avoids reliance solely on memorized steps.

To solidify our understanding, let's examine the given answer choices and see why the other options are incorrect:

A. -16 + 37i: As we calculated, this is the correct answer. The real part is -16, and the imaginary part is 37i.

B. 12 - 28i: This option is incorrect because it doesn't account for the imaginary terms resulting from the multiplication and simplification process. It seems to only consider the product of the real parts and the product of the imaginary coefficients, without properly handling the cross-terms.

C. 16 - 37i: This option has the correct magnitudes for the real and imaginary parts but the wrong signs. It's a common mistake to mix up the signs when combining terms after substituting i^2 = -1.

D. 37 + 16i: This option incorrectly swaps the real and imaginary parts and also has the wrong sign for the real part. It likely results from a misunderstanding of how to combine the terms after expanding the product.

By analyzing these options, we can see how crucial it is to follow the correct steps and avoid common errors in the multiplication and simplification process. Each choice represents a potential mistake that students might make, highlighting the importance of a thorough understanding of the process. The comparison of the correct answer with the incorrect choices reinforces the understanding of the correct procedure and helps in identifying potential pitfalls. This analytical approach not only aids in solving the immediate problem but also enhances the ability to recognize and avoid errors in similar problems. Furthermore, it develops a critical thinking approach towards mathematical problem-solving, which is essential for advanced studies in mathematics and related fields.

In conclusion, the expression equivalent to (4 + 7i)(3 + 4i) is -16 + 37i, which corresponds to answer choice A. This result is obtained by carefully applying the distributive property, simplifying the expression using the fact that i^2 = -1, and combining like terms. Understanding the multiplication of complex numbers is a fundamental skill in mathematics with applications in various fields. Avoiding common mistakes and practicing regularly are key to mastering this concept. The multiplication of complex numbers is a crucial skill in mathematics, essential for various applications in physics, engineering, and computer science. By understanding the step-by-step process of multiplying complex numbers, students can confidently solve problems and apply these concepts in more advanced contexts. The key to success lies in the meticulous application of the distributive property, the correct simplification of i^2, and the careful combination of like terms. Consistent practice and a thorough understanding of the underlying principles will lead to mastery of this important mathematical skill.