Equivalent Expressions Simplifying Complex Numbers In Mathematics

In the realm of mathematics, complex numbers present a fascinating blend of real and imaginary components, often leading to expressions that require careful simplification. Our main focus in this article is to simplify complex mathematical expressions. This article delves into the intricacies of simplifying such expressions, specifically addressing the problem: $(-\sqrt{9}+\sqrt{-4})-(2 \sqrt{576}+\sqrt{-64})$. Through a step-by-step approach, we will dissect the expression, identify equivalent forms, and clarify the underlying principles of complex number arithmetic. Our discussion will revolve around several key concepts, including the extraction of square roots of negative numbers, the properties of imaginary units, and the rules for addition and subtraction of complex numbers. By mastering these fundamentals, you'll be well-equipped to tackle a wide array of problems involving complex expressions. We aim to provide a comprehensive guide that not only answers the question at hand but also equips you with the knowledge and skills to confidently navigate the world of complex numbers.

Breaking Down the Initial Expression

Let's begin by meticulously breaking down the given expression: $(-\sqrt{9}+\sqrt{-4})-(2 \sqrt{576}+\sqrt{-64})$. This expression involves several components, each requiring careful consideration. First, we have the square roots of both positive and negative numbers. The square root of a positive number, such as $\sqrt{9}$, is straightforward. However, the square root of a negative number, such as $\sqrt{-4}$ and $\sqrt{-64}$, introduces the concept of imaginary numbers. An imaginary number is a multiple of the imaginary unit 'i', where $i = \sqrt{-1}$. Understanding this fundamental concept is crucial for simplifying the expression. Next, we encounter the term $2 \sqrt{576}$. This requires us to find the square root of 576 and then multiply the result by 2. Lastly, we have a subtraction operation between two complex numbers. To perform this subtraction, we need to remember to distribute the negative sign properly. Our goal in this section is to dissect each component of the expression, laying the groundwork for a systematic simplification process. By carefully examining each term and operation, we can pave the way for identifying equivalent expressions. Let's start by tackling the square roots and imaginary units, setting the stage for a comprehensive solution.

Simplifying Square Roots and Imaginary Units

To effectively simplify the expression $(-\sqrt{9}+\sqrt{-4})-(2 \sqrt{576}+\sqrt{-64})$, we must first address the square roots and imaginary units. The square root of 9, denoted as $\sqrt{9}$, is simply 3. This is because 3 multiplied by itself equals 9. Next, we encounter $\sqrt{-4}$. Since we cannot find a real number that, when multiplied by itself, equals -4, we introduce the concept of imaginary numbers. Recall that $i = \sqrt{-1}$. Therefore, $\sqrt{-4}$ can be rewritten as $\sqrt{4 \times -1}$, which is equivalent to $\sqrt{4} \times \sqrt{-1}$, resulting in $2i$. Similarly, let's tackle $\sqrt{-64}$. Following the same principle, we rewrite it as $\sqrt{64 \times -1}$, which is equivalent to $\sqrt{64} \times \sqrt{-1}$, giving us $8i$. Now, we need to find the square root of 576, denoted as $\sqrt{576}$. Through calculation or using a calculator, we find that the square root of 576 is 24. Therefore, $2 \sqrt{576}$ becomes $2 \times 24$, which equals 48. By carefully simplifying each square root and imaginary unit, we have transformed the initial expression into a more manageable form. We have successfully converted the square roots of negative numbers into imaginary units and calculated the square root of 576. This step is crucial for combining like terms and arriving at the final simplified expression. In the following sections, we will delve into substituting these simplified values back into the original expression and performing the necessary arithmetic operations.

Substituting and Combining Terms

Now that we have simplified the individual components, let's substitute these values back into the original expression: $(-\sqrt{9}+\sqrt{-4})-(2 \sqrt{576}+\sqrt{-64})$. We found that $\sqrt{9} = 3$, $\sqrt{-4} = 2i$, $\sqrt{576} = 24$, and $\sqrt{-64} = 8i$. Substituting these values, the expression becomes: $(-3 + 2i) - (2 \times 24 + 8i)$. Next, we simplify the term inside the second parenthesis: $2 \times 24 = 48$. So, the expression now reads: $(-3 + 2i) - (48 + 8i)$. The next step involves distributing the negative sign in front of the second parenthesis. This means we change the sign of each term inside the parenthesis: $-3 + 2i - 48 - 8i$. Now, we can combine like terms. Like terms are those that have the same variable or no variable at all. In this case, we have the real numbers -3 and -48, and the imaginary terms $2i$ and $-8i$. Combining the real numbers, we have $-3 - 48 = -51$. Combining the imaginary terms, we have $2i - 8i = -6i$. Therefore, the simplified expression is $-51 - 6i$. This process of substitution and combining like terms is fundamental to simplifying complex mathematical expressions. By carefully tracking each step and ensuring accurate arithmetic, we can transform a complicated expression into its simplest form. In the next section, we will analyze the provided answer choices to identify which ones are equivalent to our simplified expression.

Identifying Equivalent Expressions

Having simplified the original expression to $-51 - 6i$, our next crucial step is to compare this result with the provided answer choices and identify the equivalent expressions. The answer choices are:

A. $-51 + 6i$ B. $-3 + 2i + 2(24) + 8i$ C. $45 + 10i$ D. $-3 + 2i - 2(24) - 8i$ E. $-3 - 2i - 2(24) + 8i$ F. $-51 - 6i$

Let's analyze each option systematically:

• Option A: $-51 + 6i$. This expression has the correct real part (-51) but the imaginary part has the opposite sign (+6i instead of -6i). Therefore, option A is not equivalent.
• Option B: $-3 + 2i + 2(24) + 8i$. First, simplify $2(24)$ to 48. The expression becomes $-3 + 2i + 48 + 8i$. Combining like terms, we get $45 + 10i$. This is clearly different from $-51 - 6i$, so option B is not equivalent.
• Option C: $45 + 10i$. As we saw in the analysis of option B, this expression simplifies to $45 + 10i$, which is not equivalent to $-51 - 6i$.
• Option D: $-3 + 2i - 2(24) - 8i$. Simplify $2(24)$ to 48. The expression becomes $-3 + 2i - 48 - 8i$. Combining like terms, we get $-51 - 6i$. This matches our simplified expression, so option D is equivalent.
• Option E: $-3 - 2i - 2(24) + 8i$. Simplify $2(24)$ to 48. The expression becomes $-3 - 2i - 48 + 8i$. Combining like terms, we get $-51 + 6i$. This is not equivalent to $-51 - 6i$ due to the sign of the imaginary part.
• Option F: $-51 - 6i$. This is exactly the same as our simplified expression, so option F is equivalent.

Therefore, the equivalent expressions are options D and F. This methodical approach of comparing each answer choice with our simplified result ensures accuracy and reinforces the importance of careful simplification and term manipulation.

Conclusion: Mastering Complex Expression Simplification

In conclusion, simplifying complex expressions like $(-\sqrt{9}+\sqrt{-4})-(2 \sqrt{576}+\sqrt{-64})$ requires a systematic approach that involves breaking down the expression into smaller, manageable parts, simplifying each component, and then combining like terms. Our journey through this problem has highlighted several key concepts in complex number arithmetic. We began by understanding the role of imaginary units and how to extract square roots of negative numbers. We then applied these principles to simplify individual terms within the expression, such as $\sqrt{-4}$ and $\sqrt{-64}$. The process of substitution was crucial, allowing us to replace the original terms with their simplified counterparts. This step paved the way for combining like terms, both real and imaginary, ultimately leading us to the simplified form of the expression, which is $-51 - 6i$. Finally, we meticulously compared this result with the provided answer choices, identifying the equivalent expressions as options D and F. This methodical approach not only provides the correct answer but also reinforces a deeper understanding of complex number operations. Mastering these techniques is essential for success in various mathematical contexts, including algebra, calculus, and beyond. By practicing these skills and applying them to different problems, you can build confidence and proficiency in simplifying complex expressions. Remember, the key is to be methodical, pay close attention to detail, and break down complex problems into simpler steps. With consistent effort and a solid grasp of the fundamental principles, you can confidently tackle any challenge in the realm of complex numbers.

Which of the following expressions are equivalent to $(-\sqrt{9}+\sqrt{-4})-(2 \sqrt{576}+\sqrt{-64})$? Select all that apply.

Equivalent Expressions Simplifying Complex Numbers in Mathematics