Equivalent Expression Of (-√9 √-4)-(2 √576+√-64) Solution And Explanation

In the realm of mathematics, simplifying complex expressions is a fundamental skill. Often, what appears daunting at first glance can be broken down into manageable steps, revealing an elegant solution. This article delves into a specific mathematical expression, providing a detailed, step-by-step approach to finding its equivalent form. We will explore the concepts of square roots, imaginary numbers, and arithmetic operations, ultimately leading to the correct answer. Whether you're a student grappling with algebra or a math enthusiast seeking to sharpen your skills, this guide will illuminate the process of expression simplification.

Understanding the Problem: A Deep Dive into the Expression

At the heart of our mathematical journey lies the expression: $(-\sqrt{9} \sqrt{-4})-(2 \sqrt{576}+\sqrt{-64})$. This expression combines several mathematical concepts, including square roots, negative numbers under radicals (leading to imaginary numbers), and arithmetic operations such as multiplication, addition, and subtraction. To effectively tackle this problem, we need to break it down into smaller, more manageable parts. The first step involves understanding the properties of square roots and how they interact with negative numbers. Remember, the square root of a negative number is an imaginary number, denoted by the symbol 'i', where i is defined as the square root of -1. This understanding is crucial for simplifying the terms involving $\sqrt{-4}$ and $\sqrt{-64}$. Next, we need to simplify the square roots of perfect squares, such as $\sqrt{9}$ and $\sqrt{576}$. These are straightforward calculations that yield whole numbers. Once we've simplified the individual terms, we can then perform the arithmetic operations, paying close attention to the order of operations (PEMDAS/BODMAS). This involves first dealing with the multiplication within the parentheses and then handling the subtraction between the two main terms. By systematically addressing each component of the expression, we can unravel its complexity and arrive at the equivalent form. This process not only helps in solving this particular problem but also builds a strong foundation for tackling more complex mathematical challenges in the future. The key is to approach each step with clarity and precision, ensuring that no detail is overlooked. In the subsequent sections, we will delve into each of these steps with meticulous detail, providing explanations and justifications along the way.

Step-by-Step Simplification: A Journey Through the Solution

Now, let's embark on the journey of simplifying the expression $(-\sqrt{9} \sqrt{-4})-(2 \sqrt{576}+\sqrt{-64})$ step-by-step. This methodical approach will ensure clarity and accuracy in our solution. First, we'll address the square roots. We know that $\sqrt{9} = 3$, as 3 multiplied by itself equals 9. Next, we encounter $\sqrt{-4}$. Since we're dealing with the square root of a negative number, we introduce the imaginary unit 'i'. Recall that $i = \sqrt{-1}$. Therefore, $\sqrt{-4}$ can be expressed as $\sqrt{4} \cdot \sqrt{-1} = 2i$. Similarly, for $\sqrt{-64}$, we can write it as $\sqrt{64} \cdot \sqrt{-1} = 8i$. Now, let's tackle $\sqrt{576}$. This is the square root of a perfect square, and its value is 24. With these individual square roots simplified, we can substitute them back into the original expression. This gives us $(-3 \cdot 2i) - (2 \cdot 24 + 8i)$. Next, we perform the multiplications within the parentheses. $-3 \cdot 2i$ equals $-6i$, and $2 \cdot 24$ equals 48. Our expression now looks like $-6i - (48 + 8i)$. The final step involves distributing the negative sign and combining like terms. Distributing the negative sign, we get $-6i - 48 - 8i$. Combining the imaginary terms, $-6i - 8i$ equals $-14i$. Therefore, our simplified expression is $-48 - 14i$. This step-by-step simplification process highlights the importance of breaking down complex problems into smaller, more manageable parts. By carefully addressing each component and applying the correct mathematical principles, we can arrive at the solution with confidence. The journey through the simplification process not only provides the answer but also reinforces our understanding of the underlying mathematical concepts.

Identifying the Equivalent Expression: Matching the Solution

Having meticulously simplified the given expression, we've arrived at the equivalent form: $-48 - 14i$. Now, the crucial step is to identify which of the provided options matches our solution. Let's revisit the options:

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

To determine the correct match, we need to simplify each option and compare it to our solution, $-48 - 14i$. Let's analyze each option:

• Option A: $-3+2 i+2(24)+8 i$. Simplifying, we get $-3 + 2i + 48 + 8i$. Combining like terms, this becomes $45 + 10i$, which does not match our solution.
• Option B: $45+10 i$. As we saw in the analysis of Option A, this expression simplifies to $45 + 10i$, which is not our solution.
• Option C: $-51-6 i$. This expression is already in its simplest form and does not match our solution of $-48 - 14i$.
• Option D: $-51+6 i$. Similar to Option C, this expression is in its simplest form and does not match our solution.
• Option E: $-3-2 i -2(24)+8 i$. Simplifying, we get $-3 - 2i - 48 + 8i$. Combining like terms, this becomes $-51 + 6i$, which also does not match our solution.
• Option F: $-3+2 i -2(24)-8 i$. Simplifying, we get $-3 + 2i - 48 - 8i$. Combining like terms, this becomes $-51 - 6i$. This also does not match the target.

Upon careful examination, it appears there might be a discrepancy between our derived solution and the provided options. Our step-by-step simplification led us to $-48 - 14i$. Let's meticulously review our calculations to ensure accuracy. We had $(-\sqrt{9} \sqrt{-4})-(2 \sqrt{576}+\sqrt{-64})$. This became $(-3 \cdot 2i) - (2 \cdot 24 + 8i)$, then $-6i - (48 + 8i)$, and finally $-6i - 48 - 8i$, which simplified to $-48 - 14i$. If we have made no mistakes, then it might be beneficial to double-check the original problem statement and the answer choices for any potential errors or misprints. It's crucial in mathematics to ensure that the problem is correctly transcribed and that the answer choices are accurately presented. If everything is correct, it would be appropriate to conclude that none of the provided options are equivalent to the given expression. This careful process of elimination and verification underscores the importance of precision and attention to detail in mathematical problem-solving.

However, a closer look reveals a potential misinterpretation. Let's re-evaluate the initial simplification:

$(-\sqrt{9} \sqrt{-4})-(2 \sqrt{576}+\sqrt{-64})$

$(-3 * 2i) - (2 * 24 + 8i)$

$-6i - (48 + 8i)$

$-6i - 48 - 8i$

$-48 -14i$

Reviewing the options again:

A. $-3+2 i+2(24)+8 i = -3 + 2i + 48 + 8i = 45 + 10i$ B. $45+10 i$ C. $-51-6 i$ D. $-51+6 i$ E. $-3-2 i -2(24)+8 i = -3 - 2i - 48 + 8i = -51 + 6i$ F. $-3+2 i -2(24)-8 i = -3 + 2i - 48 - 8i = -51 - 6i$

Still, no match is apparent. Let's meticulously go through each step once more to ensure complete accuracy and leave no room for error. It's essential to approach mathematical problems with a systematic mindset, verifying each step to build a solid foundation for the solution.

Potential Errors and Troubleshooting: Ensuring Accuracy

In the pursuit of mathematical accuracy, it's crucial to consider potential sources of error and implement troubleshooting strategies. When faced with a discrepancy between our derived solution and the provided options, as we've encountered in this case, a systematic approach is paramount. The first step in troubleshooting is to meticulously review each step of our simplification process. This involves revisiting the initial expression, $(-\sqrt{9} \sqrt{-4})-(2 \sqrt{576}+\sqrt{-64})$, and carefully re-evaluating each simplification. We need to double-check the square root calculations, the handling of imaginary numbers, and the arithmetic operations. Did we correctly identify $\sqrt{9}$ as 3, $\sqrt{-4}$ as 2i, $\sqrt{576}$ as 24, and $\sqrt{-64}$ as 8i? Were the multiplications and additions performed accurately? Did we correctly distribute the negative sign? These are all critical questions to ask during the review process. Another potential source of error lies in the transcription of the problem itself. It's possible that we may have miscopied the original expression or one of the answer choices. Therefore, it's essential to carefully compare our working notes with the original problem statement to ensure that everything matches. If a mistake is found, it should be corrected immediately, and the simplification process should be restarted from that point. Furthermore, it's beneficial to consider alternative approaches to solving the problem. Sometimes, a fresh perspective can reveal an error that was previously overlooked. For example, we could try simplifying the expression using a different order of operations or by grouping terms differently. If, after a thorough review and re-evaluation, we still find a discrepancy, it may be necessary to consider the possibility that there is an error in the problem statement or the answer choices themselves. In such cases, it's appropriate to seek clarification from the source of the problem or to consult with a mathematical expert. The key takeaway is that accuracy in mathematics requires diligence, attention to detail, and a willingness to troubleshoot potential errors. By systematically addressing each step and considering alternative perspectives, we can increase our confidence in the correctness of our solutions.

Conclusion: The Importance of Precision in Mathematics

In conclusion, the exercise of simplifying the expression $(-\sqrt{9} \sqrt{-4})-(2 \sqrt{576}+\sqrt{-64})$ underscores the paramount importance of precision in mathematics. Our journey through this problem has highlighted the need for a systematic approach, meticulous attention to detail, and a willingness to troubleshoot potential errors. We began by breaking down the complex expression into smaller, more manageable parts, focusing on simplifying square roots and handling imaginary numbers. We then meticulously performed the arithmetic operations, adhering to the order of operations to ensure accuracy. Along the way, we encountered a discrepancy between our derived solution and the provided options, prompting a thorough review of our calculations. This process of troubleshooting reinforced the value of double-checking each step and considering alternative perspectives. While the initial discrepancy remained, the rigorous process of verification ultimately led to a deeper understanding of the problem and the potential for errors in problem statements or answer choices. This experience highlights that mathematics is not just about finding the right answer; it's about the process of logical reasoning, critical thinking, and the pursuit of accuracy. It's about developing a mindset that values precision and a willingness to question and verify every step. Whether you're a student tackling algebraic expressions or a professional applying mathematical principles in your field, the lessons learned from this exercise are invaluable. By embracing precision, fostering critical thinking, and remaining persistent in the face of challenges, you can unlock the power of mathematics and confidently navigate complex problems. The quest for mathematical understanding is a journey that requires patience, dedication, and a commitment to accuracy. And as we've seen in this example, the journey itself can be as valuable as the destination.