Simplifying The Expression 2√27 + √12 - 3√3 - 2√12
Introduction
In the realm of mathematics, simplifying expressions is a fundamental skill that allows us to present complex equations in a more manageable and understandable form. This often involves reducing terms, combining like radicals, and applying the order of operations. In this article, we will delve into the process of simplifying a given expression that involves square roots, namely: 2√27 + √12 - 3√3 - 2√12. This expression combines various square roots, some of which can be further simplified by factoring out perfect squares. Our goal is to break down each term, identify common radicals, and then combine them to arrive at the simplest form of the expression. This exercise is not only valuable for students learning algebra but also for anyone who needs to manipulate mathematical expressions in practical contexts.
Simplifying radical expressions is a key concept in algebra, offering a pathway to rewrite complex expressions in a more concise and understandable manner. The expression we aim to simplify, 2√27 + √12 - 3√3 - 2√12, presents an excellent opportunity to apply these principles. Radicals, often seen as intimidating, can be tamed by recognizing the underlying structure of their radicands – the numbers under the square root sign. The process begins with identifying perfect square factors within these radicands, which allows us to extract these factors from under the radical, thereby simplifying the overall expression. This approach not only reduces the complexity of individual terms but also paves the way for combining like terms, a crucial step in the simplification process. Our journey through this expression will involve breaking down each term into its simplest radical form, identifying terms with the same radical component, and then performing the necessary arithmetic to achieve the final simplified expression. Understanding the mechanics of simplifying radicals not only enhances one's algebraic skills but also provides a foundational understanding for more advanced mathematical concepts.
The ability to simplify expressions involving square roots is a crucial skill in mathematics, serving as a cornerstone for more advanced topics. When faced with an expression like 2√27 + √12 - 3√3 - 2√12, the initial challenge lies in recognizing the potential for simplification within the radicals. Each term in this expression contains a square root, and the key to simplifying these terms lies in identifying perfect square factors within the radicands. For instance, √27 can be rewritten as √(9 * 3), where 9 is a perfect square. Similarly, √12 can be expressed as √(4 * 3), with 4 being a perfect square. By extracting these perfect squares from under the radical sign, we can reduce the complexity of the terms and reveal underlying common factors. This process not only makes the expression more manageable but also allows us to combine like terms, which is a fundamental step in simplifying algebraic expressions. Simplifying radical expressions is not just about arriving at a final answer; it's about developing a deeper understanding of mathematical structures and fostering the ability to manipulate them effectively. In the following sections, we will systematically break down each term in the expression, simplify the radicals, and then combine like terms to arrive at the simplest form of the expression.
Step-by-Step Simplification
1. Simplify Individual Radicals
To begin, let's break down each radical term individually. The expression we're working with is 2√27 + √12 - 3√3 - 2√12. The first term, 2√27, can be simplified by recognizing that 27 is 9 times 3, and 9 is a perfect square. Thus, √27 can be rewritten as √(9 * 3), which is equal to √9 * √3, or 3√3. Multiplying this by the coefficient 2 gives us 2 * 3√3 = 6√3. The second term, √12, can be simplified similarly. 12 is 4 times 3, where 4 is a perfect square. Therefore, √12 can be rewritten as √(4 * 3) = √4 * √3 = 2√3. The third term, -3√3, is already in its simplest form, as 3 is a prime number and cannot be factored into a perfect square. The fourth term, -2√12, can be simplified using our previous simplification of √12. We know that √12 = 2√3, so -2√12 becomes -2 * 2√3 = -4√3. By simplifying each radical term separately, we've laid the groundwork for combining like terms, which will lead us to the final simplified form of the expression. This step highlights the importance of recognizing perfect square factors within radicals, a crucial skill in simplifying radical expressions.
The initial step in simplifying the expression 2√27 + √12 - 3√3 - 2√12 involves dissecting each radical term and identifying any perfect square factors within the radicand. This process is akin to peeling back layers to reveal the underlying simplicity. Consider the term 2√27. The radicand, 27, can be factored into 9 and 3, where 9 is a perfect square. This allows us to rewrite √27 as √(9 * 3). Applying the property of square roots that √(a * b) = √a * √b, we get √9 * √3. Since √9 equals 3, the expression simplifies to 3√3. Multiplying by the initial coefficient of 2, we have 2 * 3√3 = 6√3. Moving on to the term √12, we recognize that 12 can be factored into 4 and 3, where 4 is a perfect square. Thus, √12 becomes √(4 * 3), which simplifies to √4 * √3. Knowing that √4 equals 2, we get 2√3. The third term, -3√3, is already in its simplest form, as the radicand 3 has no perfect square factors other than 1. For the fourth term, -2√12, we can use our previous simplification of √12, which is 2√3. Therefore, -2√12 becomes -2 * 2√3 = -4√3. This meticulous breakdown of each radical term is fundamental to the simplification process, allowing us to identify like terms that can be combined in the subsequent steps. By simplifying individual radicals, we not only reduce the complexity of each term but also set the stage for a more straightforward combination of like terms.
The key to efficiently simplifying the expression 2√27 + √12 - 3√3 - 2√12 lies in mastering the art of simplifying individual radicals. This involves a systematic approach of identifying perfect square factors within the radicand and extracting them from under the square root sign. Let’s begin with 2√27. The number 27 can be expressed as the product of 9 and 3, where 9 is a perfect square. Thus, we can rewrite √27 as √(9 × 3). Using the property that the square root of a product is the product of the square roots, √(9 × 3) becomes √9 × √3. Since √9 equals 3, we simplify √27 to 3√3. Multiplying this by the coefficient 2 gives us 2 × 3√3 = 6√3. Next, we address the term √12. The radicand 12 can be factored into 4 and 3, with 4 being a perfect square. Therefore, √12 can be expressed as √(4 × 3), which is equivalent to √4 × √3. Knowing that √4 equals 2, we simplify √12 to 2√3. The term -3√3 is already in its simplest form, as the radicand 3 has no perfect square factors other than 1. Finally, we consider the term -2√12. We’ve already determined that √12 simplifies to 2√3, so -2√12 becomes -2 × 2√3 = -4√3. By meticulously simplifying each radical term in this manner, we not only make the expression more manageable but also pave the way for combining like terms. This step is crucial for reducing the expression to its simplest form, and it underscores the importance of recognizing and extracting perfect square factors from radicals.
2. Combine Like Terms
After simplifying each radical, we now have the expression 6√3 + 2√3 - 3√3 - 4√3. Notice that all the terms now have the same radical, √3. This allows us to combine them like any other like terms in algebra. To combine like terms, we simply add or subtract the coefficients of the terms. In this case, we have the coefficients 6, 2, -3, and -4. Adding these together, we get 6 + 2 - 3 - 4 = 1. Therefore, the simplified expression is 1√3, which is more commonly written as √3. This step demonstrates the power of simplifying individual terms before attempting to combine them. By reducing each radical to its simplest form, we made it clear that all the terms were indeed like terms, allowing for a straightforward combination. This process not only simplifies the expression but also reduces the likelihood of errors in calculation. The final result, √3, is the simplest form of the original expression, showcasing the effectiveness of simplifying radicals and combining like terms.
With the individual radicals simplified, the expression now stands as 6√3 + 2√3 - 3√3 - 4√3. The beauty of this transformation lies in the uniformity of the radical component – each term now contains √3. This common thread allows us to treat these terms as like terms, similar to how we combine variables in algebraic expressions. The process of combining like terms involves summing or subtracting the coefficients while keeping the radical component unchanged. In our expression, the coefficients are 6, 2, -3, and -4. To combine these, we perform the arithmetic operation: 6 + 2 - 3 - 4. Adding 6 and 2 gives us 8, then subtracting 3 yields 5, and finally, subtracting 4 leaves us with 1. Thus, the combined coefficient is 1. Attaching the radical component √3 to this coefficient, we arrive at the simplified term 1√3. While mathematically correct, it is conventional to omit the coefficient 1 when it is present, simplifying the expression further to √3. This final simplification represents the culmination of our efforts, showcasing the power of simplifying radicals and combining like terms to reduce a complex expression to its most basic form. The result, √3, is not only concise but also clearly represents the value of the original expression in its simplest radical form.
Having simplified the individual radicals, the expression now takes the form 6√3 + 2√3 - 3√3 - 4√3. The significance of this transformation is that all terms now share a common radical, namely √3. This crucial step allows us to combine the terms as like terms, a process akin to combining variables in algebraic expressions. The mechanics of combining like terms involve focusing on the coefficients of each term while keeping the radical part constant. In this expression, the coefficients are 6, 2, -3, and -4. We proceed to perform the arithmetic operation on these coefficients: 6 + 2 - 3 - 4. First, adding 6 and 2 yields 8. Then, subtracting 3 from 8 gives us 5. Finally, subtracting 4 from 5 results in 1. Therefore, the sum of the coefficients is 1. This means that the simplified expression is 1√3. However, in mathematical convention, we typically omit the coefficient 1 when it is multiplied by a radical. Thus, 1√3 is more simply written as √3. This final form, √3, represents the simplest form of the original expression. The process of simplifying and combining like terms has allowed us to reduce a seemingly complex expression into a single, elegant radical term. This result not only highlights the efficiency of these simplification techniques but also underscores the importance of recognizing and leveraging common factors within mathematical expressions.
Conclusion
In conclusion, the simplified form of the expression 2√27 + √12 - 3√3 - 2√12 is √3. This simplification process involved two key steps: first, simplifying individual radicals by identifying and extracting perfect square factors, and second, combining like terms by adding or subtracting their coefficients. The ability to simplify radical expressions is a fundamental skill in algebra and is essential for solving more complex mathematical problems. This exercise demonstrates the importance of breaking down complex problems into smaller, manageable steps and applying the rules of algebra to arrive at a solution. Understanding how to simplify radicals not only enhances one's mathematical proficiency but also provides a foundation for more advanced topics in mathematics. The journey from the initial expression to the simplified form √3 showcases the power of algebraic manipulation and the elegance of mathematical simplification.
To summarize, the journey of simplifying the expression 2√27 + √12 - 3√3 - 2√12 has illuminated the critical role of simplifying radicals and combining like terms in algebraic manipulations. We embarked on this process by first simplifying each radical term individually. This involved identifying perfect square factors within the radicands and extracting them from under the square root sign. For instance, √27 was simplified to 3√3, and √12 was simplified to 2√3. This initial step not only reduced the complexity of each term but also revealed a common radical component, √3, across all terms. The subsequent step involved combining like terms, which was made possible by the shared radical. By adding and subtracting the coefficients of the terms with the same radical, we were able to consolidate the expression. The arithmetic operation 6 + 2 - 3 - 4 resulted in a coefficient of 1, leading to the simplified expression 1√3, which is conventionally written as √3. This final form represents the culmination of our efforts, showcasing the elegance and efficiency of algebraic simplification techniques. The transformation from the original complex expression to the concise form √3 underscores the importance of mastering radical simplification and the combination of like terms. These skills are not only fundamental to algebra but also serve as building blocks for more advanced mathematical concepts.
The simplification of the expression 2√27 + √12 - 3√3 - 2√12 exemplifies the core principles of algebraic manipulation, highlighting the significance of simplifying radicals and combining like terms. The process began with a methodical breakdown of each term, focusing on the simplification of individual radicals. By recognizing and extracting perfect square factors from the radicands, we were able to reduce the complexity of each term significantly. √27, for example, was transformed into 3√3, and √12 was simplified to 2√3. This initial step was crucial in revealing the common radical component, √3, across all terms in the expression. With the simplified radicals in place, the next step was to combine like terms. This involved adding and subtracting the coefficients of the terms that shared the same radical. The coefficients 6, 2, -3, and -4 were combined, resulting in a sum of 1. This led to the simplified expression 1√3, which is conventionally expressed as √3. The journey from the original expression to the final simplified form, √3, underscores the power of algebraic techniques in reducing complex expressions to their most basic components. The ability to simplify radicals and combine like terms is not only a valuable skill in algebra but also a foundational element for understanding and solving more advanced mathematical problems. The final result, √3, is a testament to the elegance and efficiency of these simplification methods.
