Simplifying $(\sqrt{10}-2 \sqrt{3})^2$ A Step-by-Step Guide

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Introduction

In this comprehensive exploration, we will delve into the intricacies of simplifying the expression (10−23)2(\sqrt{10}-2 \sqrt{3})^2. This seemingly straightforward mathematical problem opens the door to a fascinating journey through the realms of algebraic manipulation, radical simplification, and the elegant application of fundamental mathematical principles. Our discussion aims to not only provide a step-by-step solution but also to illuminate the underlying concepts that make this problem a valuable exercise in mathematical understanding. We will break down the expression, apply the appropriate algebraic identities, and meticulously simplify the result, ensuring clarity and precision at each stage. Whether you are a student seeking to enhance your algebra skills or a mathematics enthusiast eager to explore the beauty of mathematical simplification, this detailed analysis will offer valuable insights and a deeper appreciation for the power of mathematical tools.

At the heart of this problem lies the concept of expanding a squared binomial. The expression (10−23)2(\sqrt{10}-2 \sqrt{3})^2 is a perfect example of this, and to simplify it effectively, we must recall and apply the algebraic identity (a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab + b^2. This identity is a cornerstone of algebra and provides a systematic way to expand expressions of this form. By correctly identifying 'a' and 'b' in our expression and substituting them into the identity, we can transform the problem into a more manageable form. This initial step is crucial, as it sets the stage for the subsequent simplification of radical terms and the combination of like terms. The beauty of this approach lies in its methodical nature, allowing us to break down a complex expression into simpler, more digestible components.

Furthermore, the simplification process involves a thorough understanding of radical properties. Radicals, or square roots, often require careful manipulation to arrive at their simplest form. In our case, we will encounter terms involving square roots of 10 and 3, which will necessitate the application of properties such as aâ‹…b=ab\sqrt{a} \cdot \sqrt{b} = \sqrt{ab}. This property allows us to combine and simplify radical expressions, ultimately leading to a more concise and elegant solution. Additionally, we will need to address terms where a radical is squared, such as (10)2(\sqrt{10})^2, which simply evaluates to 10. Mastering these radical simplification techniques is essential not only for this particular problem but also for a wide range of mathematical challenges involving radicals. The ability to confidently manipulate and simplify radicals is a hallmark of strong algebraic proficiency.

Step-by-Step Simplification of (10−23)2(\sqrt{10}-2 \sqrt{3})^2

To begin, we will employ the algebraic identity (a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab + b^2. In our expression, a=10a = \sqrt{10} and b=23b = 2\sqrt{3}. Substituting these values into the identity, we get:

(10−23)2=(10)2−2(10)(23)+(23)2(\sqrt{10}-2 \sqrt{3})^2 = (\sqrt{10})^2 - 2(\sqrt{10})(2\sqrt{3}) + (2\sqrt{3})^2

Now, let's simplify each term individually. The first term, (10)2(\sqrt{10})^2, is simply 10, as squaring a square root cancels out the radical. The third term, (23)2(2\sqrt{3})^2, requires us to square both the coefficient and the radical. Squaring 2 gives us 4, and squaring 3\sqrt{3} gives us 3. Therefore, (23)2=4⋅3=12(2\sqrt{3})^2 = 4 \cdot 3 = 12. The middle term, −2(10)(23)- 2(\sqrt{10})(2\sqrt{3}), involves multiplying the coefficients and the radicals separately. Multiplying the coefficients, we have −2⋅2=−4-2 \cdot 2 = -4. Multiplying the radicals, we have 10⋅3=30\sqrt{10} \cdot \sqrt{3} = \sqrt{30}. Thus, the middle term simplifies to −430-4\sqrt{30}.

Substituting these simplified terms back into our expression, we have:

(10−23)2=10−430+12(\sqrt{10}-2 \sqrt{3})^2 = 10 - 4\sqrt{30} + 12

Finally, we combine the constant terms, 10 and 12, to obtain:

(10−23)2=22−430(\sqrt{10}-2 \sqrt{3})^2 = 22 - 4\sqrt{30}

This is the simplified form of the expression. It's important to note that 30\sqrt{30} cannot be simplified further, as 30 does not have any perfect square factors other than 1. Therefore, our final answer, 22−43022 - 4\sqrt{30}, represents the most concise and accurate simplification of the original expression.

This step-by-step process highlights the importance of breaking down a complex problem into smaller, more manageable steps. By carefully applying the algebraic identity and simplifying each term individually, we were able to arrive at the final answer with confidence. This approach not only ensures accuracy but also enhances our understanding of the underlying mathematical principles at play. The ability to systematically simplify expressions is a crucial skill in mathematics, and this example serves as a valuable illustration of how to effectively tackle such problems.

Key Concepts and Techniques

Throughout the simplification process, we have employed several key concepts and techniques that are fundamental to algebraic manipulation. One of the most important concepts is the algebraic identity (a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab + b^2. This identity provides a direct and efficient way to expand squared binomials, saving us the effort of manually multiplying the expression by itself. Understanding and memorizing this identity is crucial for simplifying a wide range of algebraic expressions. In our case, it allowed us to transform the original expression into a form that was much easier to simplify.

Another essential technique is the simplification of radicals. Radicals, particularly square roots, often require careful manipulation to express them in their simplest form. We utilized the property aâ‹…b=ab\sqrt{a} \cdot \sqrt{b} = \sqrt{ab} to combine radicals and simplify terms. Additionally, we recognized that (x)2=x(\sqrt{x})^2 = x, which allowed us to eliminate square roots when squaring radical terms. These techniques are essential for working with radicals and are applicable in various mathematical contexts. The ability to confidently manipulate radicals is a hallmark of strong algebraic skills.

Furthermore, the process of combining like terms is a fundamental aspect of algebraic simplification. In our example, we combined the constant terms 10 and 12 to arrive at the final constant term of 22. This step is crucial for presenting the expression in its most concise and readable form. Combining like terms ensures that we are only left with distinct terms that cannot be further simplified. This technique is not only applicable to constant terms but also to terms involving variables or radicals.

Finally, the importance of a systematic approach cannot be overstated. By breaking down the problem into smaller, more manageable steps, we were able to simplify the expression accurately and efficiently. This approach involves carefully applying the algebraic identity, simplifying each term individually, and then combining like terms. A systematic approach not only minimizes the risk of errors but also enhances our understanding of the underlying mathematical principles. It allows us to tackle complex problems with confidence and clarity.

Conclusion

In conclusion, the simplification of the expression (10−23)2(\sqrt{10}-2 \sqrt{3})^2 provides a valuable exercise in algebraic manipulation and radical simplification. By applying the algebraic identity (a−b)2=a2−2ab+b2(a - b)^2 = a^2 - 2ab + b^2 and employing techniques for simplifying radicals, we were able to transform the expression into its simplest form: 22−43022 - 4\sqrt{30}. This process highlights the importance of understanding fundamental algebraic principles and mastering the techniques necessary for manipulating radicals. The ability to confidently simplify expressions is a crucial skill in mathematics, and this example serves as a testament to the power of systematic problem-solving.

Throughout our discussion, we have emphasized the significance of breaking down complex problems into smaller, more manageable steps. This approach not only ensures accuracy but also enhances our understanding of the underlying mathematical concepts. By carefully applying the appropriate identities and techniques, we can tackle a wide range of mathematical challenges with confidence and clarity. The journey from the initial expression to the final simplified form showcases the elegance and precision of mathematical reasoning. The result, 22−43022 - 4\sqrt{30}, stands as a testament to the beauty of mathematical simplification and the power of algebraic tools.