Simplifying Radicals A Step-by-Step Guide To √6(2√10 + 3√2)
In mathematics, simplifying expressions is a fundamental skill that allows us to represent complex equations in a more manageable and understandable form. This article focuses on simplifying the expression √6(2√10 + 3√2), a task that involves the application of several key concepts in algebra and radical simplification. Understanding these concepts is crucial for anyone studying mathematics, as they form the basis for more advanced topics. The process involves distributing the radical, simplifying individual terms, and combining like terms to arrive at the simplest form of the expression.
Understanding the Basics of Radicals
Before diving into the simplification process, it's essential to understand what radicals are and how they behave. A radical, often represented by the square root symbol (√), indicates a root of a number. For instance, √9 represents the square root of 9, which is 3, because 3 * 3 = 9. Radicals can also represent cube roots, fourth roots, and so on, but in this case, we primarily deal with square roots. Radicals can be multiplied together using the property √(a) * √(b) = √(a * b), which is crucial for simplifying expressions involving radicals. To simplify a radical expression effectively, it's also important to recognize perfect squares within the radical. For example, √100 can be easily simplified to 10 because 100 is a perfect square (10 * 10 = 100). When dealing with expressions containing radicals, the goal is often to simplify them by removing perfect square factors from the radicand (the number under the radical sign). This can make the expression easier to understand and work with. The process often involves breaking down the radicand into its prime factors and then identifying pairs of identical factors. Each pair can be taken out of the radical as a single factor. Understanding these properties and techniques is essential for tackling the given expression effectively. Simplifying radicals is not just a mathematical exercise; it's a practical skill used in various fields, including physics, engineering, and computer science. By mastering the simplification of radicals, one can more easily manipulate and solve equations, making it a crucial step in mathematical proficiency.
Step-by-Step Simplification of √6(2√10 + 3√2)
To simplify the given expression, √6(2√10 + 3√2), we will follow a step-by-step approach, applying the distributive property and the rules of radical multiplication. The primary goal is to eliminate radicals where possible and combine like terms to achieve the simplest form. Each step in this process involves specific mathematical principles, ensuring that the final result is accurate and easily understood.
Step 1: Distribute √6 Across the Terms
The first step in simplifying the expression is to distribute the √6 across the terms inside the parentheses. This means we multiply √6 by both 2√10 and 3√2. Applying the distributive property, we get:
√6 * (2√10) + √6 * (3√2)
This step uses the distributive property of multiplication over addition, a fundamental concept in algebra. It allows us to break down the complex expression into smaller, more manageable parts. The key here is to ensure that each term inside the parentheses is correctly multiplied by the term outside. By distributing √6, we set the stage for further simplification by combining the radicals.
Step 2: Multiply the Radicals
Next, we multiply the radicals together. Remember that √(a) * √(b) = √(a * b). Applying this rule, we get:
2 * √(6 * 10) + 3 * √(6 * 2)
This simplifies to:
2√60 + 3√12
This step is crucial as it combines the radicals into single terms, making it easier to identify and extract perfect square factors. The multiplication under the radical sign allows us to consolidate the numbers, which is a key step in simplification. By multiplying the radicals, we are essentially combining them into a form that can be further broken down into simpler components.
Step 3: Simplify the Radicals √60 and √12
Now, we simplify the radicals √60 and √12 by identifying and extracting perfect square factors. Let's start with √60. We can break down 60 into its prime factors: 60 = 2 * 2 * 3 * 5. We see a pair of 2s, which means we can extract a 2 from the radical. Thus, √60 = √(2^2 * 3 * 5) = 2√(3 * 5) = 2√15.
Next, we simplify √12. The prime factorization of 12 is 12 = 2 * 2 * 3. Again, we have a pair of 2s, so we can extract a 2 from the radical. Thus, √12 = √(2^2 * 3) = 2√3.
Simplifying radicals involves breaking down the number under the radical into its prime factors and then identifying pairs of factors. Each pair can be taken out of the radical as a single factor. This process is based on the property that √(a^2) = a. By simplifying the radicals, we are reducing the expression to its most basic form, making it easier to understand and manipulate.
Step 4: Substitute the Simplified Radicals Back into the Expression
Now that we have simplified √60 and √12, we substitute these back into our expression:
2√60 + 3√12 becomes 2(2√15) + 3(2√3)
This substitution is a crucial step in piecing together the simplified form of the original expression. By replacing the original radicals with their simplified forms, we are moving closer to a final expression that is free of complex radicals.
Step 5: Perform the Multiplication
Next, we perform the multiplication:
2(2√15) + 3(2√3) = 4√15 + 6√3
This step involves multiplying the coefficients with the radicals. It's a straightforward application of multiplication, but it's essential to ensure that the numbers are correctly multiplied. By performing this multiplication, we are further simplifying the expression and preparing it for the final step of combining like terms.
Step 6: Check for Like Terms and Combine
Finally, we check for like terms. Like terms are terms that have the same radical part. In this case, 4√15 and 6√3 do not have the same radical part (√15 and √3 are different), so they cannot be combined. Therefore, the expression is already in its simplest form.
This final check is crucial to ensure that the expression is indeed in its simplest form. Combining like terms is a fundamental step in simplifying algebraic expressions, but it's only possible when the radical parts are the same. Since 4√15 and 6√3 cannot be combined, the simplified expression is complete.
Final Simplified Expression
Therefore, the simplified form of √6(2√10 + 3√2) is:
4√15 + 6√3
This final expression is the most simplified form of the original expression. It contains no perfect square factors under the radicals and no like terms that can be combined. The process of simplification has transformed the original expression into a more manageable and understandable form, demonstrating the power of algebraic manipulation.
Conclusion
Simplifying expressions involving radicals is a fundamental skill in mathematics. In this article, we methodically simplified the expression √6(2√10 + 3√2) by applying the distributive property, multiplying radicals, extracting perfect square factors, and combining like terms. The result, 4√15 + 6√3, represents the simplest form of the original expression. Mastering these techniques is essential for success in algebra and beyond. Understanding how to manipulate and simplify radicals not only enhances problem-solving skills but also provides a deeper insight into the structure and properties of numbers. By following a step-by-step approach and applying the rules of radicals, complex expressions can be transformed into simpler, more manageable forms. This skill is not just applicable in academic settings but also in various real-world scenarios where mathematical expressions need to be simplified for practical applications.
