Simplifying The Expression $2\sqrt{2}-6-7\sqrt{2}$ A Step-by-Step Guide

Understanding the Expression

In this mathematical discussion, we aim to simplify the expression $2\sqrt{2}-6-7\sqrt{2}$. This expression involves a combination of rational numbers and irrational numbers, specifically those involving square roots. To effectively simplify it, we need to understand the properties of these numbers and how they interact under arithmetic operations. At its core, this task involves combining like terms, which in this case are the terms involving the square root of 2. Before diving into the simplification steps, let’s explore the fundamental concepts that underpin this process. We'll begin by identifying the types of numbers involved – rational and irrational – and then discuss how these numbers behave when subjected to operations such as addition and subtraction. The goal is to present a clear, step-by-step guide that demystifies the process, making it accessible even to those who might find such expressions daunting at first glance. This detailed approach ensures a solid understanding, allowing for confidence in tackling similar problems in the future. Ultimately, simplification isn't just about finding the correct answer; it's about understanding the underlying mathematical principles that make that answer possible. Therefore, the journey through simplifying this expression will also be a journey through key mathematical concepts.

Identifying Like Terms

The cornerstone of simplifying algebraic expressions, including the one at hand, is the identification and combination of like terms. In the expression $2\sqrt{2}-6-7\sqrt{2}$, like terms are those that share the same radical part. This means they have the same number under the square root symbol (or cube root, etc., in other scenarios). Here, we have two terms that involve $\sqrt{2}$: $2\sqrt{2}$ and $-7\sqrt{2}$. The term -6, being a rational number without any radical part, stands alone and cannot be combined directly with the other two terms. Recognizing these like terms is the initial critical step, as it sets the stage for the next operation: combining these terms through addition and subtraction. This process is similar to combining variables in algebra, where you can add or subtract terms that have the same variable and exponent (e.g., 3x + 5x = 8x). The principle remains consistent; we are essentially treating $\sqrt{2}$ as a variable in this context. By focusing on the common radical part, we simplify the expression in a logical and structured manner, paving the way for an efficient and accurate solution. This method underscores the importance of pattern recognition in mathematics, as identifying like terms is a skill applicable across various algebraic manipulations.

Understanding Irrational Numbers

To further appreciate the process of simplifying $2\sqrt{2}-6-7\sqrt{2}$, a basic understanding of irrational numbers is essential. Irrational numbers are real numbers that cannot be expressed as a simple fraction, meaning they cannot be written in the form p/q, where p and q are integers. A classic example of an irrational number is the square root of 2, denoted as $\sqrt{2}$. When you encounter such numbers in expressions, they cannot be directly combined with rational numbers (like -6 in our case) unless those rational numbers are also multiplied by the same irrational number. The reason for this lies in the fundamental nature of irrational numbers; their decimal representations are non-repeating and non-terminating. This characteristic makes them distinct from rational numbers, whose decimal representations either terminate or repeat. In our expression, the presence of $\sqrt{2}$ necessitates a specific approach to simplification, namely, combining the terms that include this irrational number separately from the rational term. Grasping this distinction is crucial for accurately handling expressions with radicals and for navigating the nuances of real number arithmetic. This also highlights why understanding the number system is foundational for advanced mathematical operations.

Step-by-Step Simplification

Now that we have a solid understanding of the underlying concepts, let's proceed with the step-by-step simplification of the expression $2\sqrt{2}-6-7\sqrt{2}$. This process will involve identifying like terms, combining them, and then presenting the simplified form. The goal is to make each step clear and logical, ensuring a comprehensive understanding of the method.

Step 1: Identify Like Terms

The first step in simplifying any algebraic expression is to identify the like terms. As discussed earlier, like terms are those that share the same variable or radical part. In our expression, $2\sqrt{2}$ and $-7\sqrt{2}$ are like terms because they both contain $\sqrt{2}$. The term -6 is a constant and does not have $\sqrt{2}$, so it is not a like term with the other two. Correctly identifying like terms is crucial because it dictates which terms can be combined. This step might seem straightforward, but it is the foundation for the rest of the simplification process. If like terms are misidentified, the subsequent steps will lead to an incorrect result. Therefore, careful attention should be paid at this stage to ensure accuracy. Once like terms are correctly identified, we can proceed to the next step, which involves combining these terms through addition and subtraction.

Step 2: Combine Like Terms

After identifying the like terms in the expression $2\sqrt{2}-6-7\sqrt{2}$, the next step is to combine them. This involves performing the arithmetic operations on the coefficients of the like terms. In our case, we have $2\sqrt{2}$ and $-7\sqrt{2}$. We can combine these by adding their coefficients: 2 and -7. This gives us 2 + (-7) = -5. Therefore, combining $2\sqrt{2}$ and $-7\sqrt{2}$ results in $-5\sqrt{2}$. The term -6 remains unchanged as it is not a like term with the others. The process of combining like terms is akin to simplifying algebraic expressions involving variables. For example, just as 2x - 7x simplifies to -5x, similarly, $2\sqrt{2} - 7\sqrt{2}$ simplifies to $-5\sqrt{2}$. This analogy helps to solidify the concept and make it more relatable. By focusing on the coefficients and treating the radical part as a common factor, we efficiently reduce the complexity of the expression.

Step 3: Write the Simplified Expression

Once the like terms have been combined, the final step is to write the simplified expression. In our case, after combining $2\sqrt{2}$ and $-7\sqrt{2}$ to get $-5\sqrt{2}$, and considering the constant term -6, the simplified expression is $-5\sqrt{2} - 6$. This is the most concise form of the original expression, as no further simplification is possible. The terms $-5\sqrt{2}$ and -6 cannot be combined because one is an irrational term and the other is a rational term. Presenting the simplified expression clearly is as important as the process of simplification itself. A well-organized and clearly written final answer leaves no room for ambiguity and demonstrates a thorough understanding of the problem. This step is the culmination of all the previous steps, showcasing the result of accurate identification of like terms and their subsequent combination. The final simplified form provides a more streamlined representation of the original expression, making it easier to understand and use in further calculations or analyses.

Final Simplified Form

In conclusion, after carefully following the steps of identifying and combining like terms, the final simplified form of the expression $2\sqrt{2}-6-7\sqrt{2}$ is $-5\sqrt{2} - 6$. This result represents the most reduced and concise version of the original expression. The process involved recognizing that the terms $2\sqrt{2}$ and $-7\sqrt{2}$ were like terms because they both contained the $\sqrt{2}$ radical. These terms were then combined by performing the arithmetic operation on their coefficients (2 and -7), resulting in $-5\sqrt{2}$. The constant term -6 remained unchanged as it could not be combined with the radical term. The final expression, $-5\sqrt{2} - 6$, consists of an irrational term and a rational term, which cannot be further simplified. This underscores the importance of understanding the properties of different types of numbers and how they interact under mathematical operations. Simplifying expressions is a fundamental skill in algebra and mathematics, and this example illustrates the key steps involved in achieving a simplified result. The clarity and conciseness of the final form highlight the value of the simplification process, making the expression easier to interpret and use in subsequent calculations or problems.