In the realm of mathematics, particularly in algebra, dealing with radicals and simplifying expressions involving them is a fundamental skill. This article delves into the process of solving a specific sum involving radicals: 4(⁵√x²y) + 3(⁵√x²y). We will break down the components of the expression, explain the steps to simplify it, and arrive at the correct answer. Whether you're a student grappling with algebra or simply someone looking to refresh your math skills, this guide will provide a clear and comprehensive understanding of how to approach such problems. This exploration will not only enhance your ability to solve similar mathematical challenges but also deepen your appreciation for the elegance and precision of algebraic manipulations.
Breaking Down the Expression
To effectively tackle the given expression, 4(⁵√x²y) + 3(⁵√x²y), it's crucial to understand its components. The expression involves the addition of two terms, each containing a radical. A radical, in this case, is represented by the fifth root (⁵√) of the expression x²y. The numbers 4 and 3 are coefficients that multiply the radical term. The expression x²y itself consists of variables x and y, where x is raised to the power of 2. Understanding these components is the first step in simplifying the expression. The fifth root signifies that we are looking for a number that, when raised to the power of 5, equals the expression inside the radical, which is x²y. The coefficients 4 and 3 indicate how many times this radical term is being added. By recognizing these individual parts, we can begin to see how the expression can be simplified using algebraic principles. This foundational understanding is essential for anyone looking to master radical expressions and their manipulation. Properly dissecting the expression not only aids in simplifying it but also provides a clearer pathway for solving more complex algebraic problems involving radicals.
Identifying Like Terms
In order to simplify the expression 4(⁵√x²y) + 3(⁵√x²y), the crucial step is recognizing and grouping like terms. In algebraic terms, like terms are those that have the same variable parts raised to the same powers. In this specific expression, both terms contain the same radical, which is the fifth root of x²y (⁵√x²y). This shared radical component makes them like terms. Think of it as adding similar objects; for instance, you can easily add 4 apples and 3 apples because they are both apples. Similarly, in our expression, we have 4 of the “⁵√x²y” term and 3 of the same term. Because the radical part (⁵√x²y) is identical in both terms, we can combine them much like we combine similar items in everyday arithmetic. This identification of like terms is a cornerstone of algebraic simplification. By pinpointing these similarities, we set the stage for a straightforward addition operation, leading us closer to the simplified form of the expression. Recognizing like terms is not just a procedural step; it’s a fundamental concept that underpins much of algebraic manipulation and simplification.
Simplifying the Expression
Once we've identified that 4(⁵√x²y) and 3(⁵√x²y) are like terms, the next step is to combine them. This process is akin to adding coefficients of the same variable in simpler algebraic expressions. In this case, we add the coefficients 4 and 3, while keeping the radical term (⁵√x²y) the same. So, we perform the addition: 4 + 3 = 7. This means that when we combine the two terms, we get 7 times the radical term, which can be written as 7(⁵√x²y). This simplification is possible because the radical part (⁵√x²y) acts as a common factor. We are essentially factoring out (⁵√x²y) and adding the coefficients. The simplified expression, 7(⁵√x²y), is much cleaner and easier to work with compared to the original form. This process highlights the power of recognizing like terms and applying basic arithmetic operations to simplify algebraic expressions. The ability to simplify expressions like this is a foundational skill in algebra, enabling further problem-solving and mathematical manipulations. By condensing the original two terms into a single, simplified term, we’ve made the expression more manageable and understandable.
Arriving at the Solution
Having simplified the expression 4(⁵√x²y) + 3(⁵√x²y) to 7(⁵√x²y), we have now arrived at the solution. This final form represents the sum in its simplest terms. The coefficient 7 indicates that we have a total of 7 units of the radical term (⁵√x²y). The radical term itself, the fifth root of x²y, remains unchanged, as it is the common factor that was combined. There are no further simplifications possible without additional information or constraints on the variables x and y. The solution 7(⁵√x²y) is not only concise but also accurately reflects the result of the original addition. In the context of multiple-choice questions or further mathematical operations, this simplified form would be the preferred way to represent the sum. It eliminates any ambiguity and presents the answer in the most straightforward manner. The journey from the initial expression to the simplified solution demonstrates the elegance and efficiency of algebraic manipulation. By following the steps of identifying like terms, combining coefficients, and maintaining the radical, we've successfully navigated the problem and reached a clear and correct answer.
Conclusion
In conclusion, the sum 4(⁵√x²y) + 3(⁵√x²y) simplifies to 7(⁵√x²y). This process involves recognizing the like terms within the expression, adding their coefficients, and retaining the common radical part. This example underscores the importance of understanding the fundamental principles of algebra, particularly the manipulation of radicals and the identification of like terms. By mastering these concepts, one can confidently approach and solve a variety of algebraic problems. The ability to simplify expressions is not just a mathematical skill; it's a critical thinking exercise that enhances problem-solving capabilities in various fields. Whether you're dealing with complex equations or simple sums, the principles remain the same. A clear understanding of algebraic rules allows for efficient and accurate problem-solving. This exploration of radical expressions serves as a stepping stone for more advanced mathematical concepts, emphasizing the interconnectedness of mathematical knowledge. The simplicity and elegance of the solution, 7(⁵√x²y), highlight the beauty of algebraic simplification and its power in making complex problems manageable.
Correct Answer: C. 7(⁵√x²y)
