Identifying Like Radicals Of 3x√5 A Comprehensive Guide

Before we dive into identifying like radicals of $3x\sqrt{5}$, let's first define what like radicals are. In mathematics, like radicals are radical expressions that have the same index (the small number outside the radical sign indicating the root) and the same radicand (the expression under the radical sign). In simpler terms, like radicals are radicals that can be combined through addition or subtraction because they share the same root and the same number or expression inside the root.

To illustrate this concept, consider the expressions $2\sqrt{3}$ and $5\sqrt{3}$. These are like radicals because they both have a square root (index of 2) and the radicand is 3. Therefore, we can combine them: $2\sqrt{3} + 5\sqrt{3} = 7\sqrt{3}$. On the other hand, $2\sqrt{3}$ and $2\sqrt{5}$ are not like radicals because they have different radicands, even though they have the same index. Similarly, $2\sqrt{3}$ and $2\sqrt[3]{3}$ are not like radicals because they have different indices, even though they have the same radicand. Understanding this foundational concept is crucial for simplifying radical expressions and solving algebraic equations involving radicals. The ability to correctly identify like radicals is a fundamental skill in algebra, enabling students to simplify expressions and solve equations more efficiently. Recognizing like radicals also helps in performing operations such as addition, subtraction, multiplication, and division involving radicals. This skill is not only essential for academic success but also has practical applications in various fields, including engineering, physics, and computer science, where radical expressions often arise in mathematical models and problem-solving scenarios. Therefore, a solid grasp of like radicals is invaluable for anyone pursuing further studies or careers in STEM-related fields.

Identifying Like Radicals of $3x\sqrt{5}$

Now that we understand the definition of like radicals, let's identify the like radical to $3x\sqrt{5}$ among the given options. The expression $3x\sqrt{5}$ consists of a coefficient $3x$ multiplied by the square root of 5. To find a like radical, we need to look for an expression that also has a square root (index of 2) and a radicand of 5. The coefficient can be different, as it will simply be combined during addition or subtraction.

Consider the options provided:

• A. $x(\sqrt[3]{5})$
• B. $\sqrt{5y}$
• C. $3(\sqrt[3]{5x})$
• D. $y\sqrt{5}$

Let's analyze each option step by step. Option A, $x(\sqrt[3]{5})$, has an index of 3, indicating a cube root, while our target expression $3x\sqrt{5}$ has an index of 2, representing a square root. Since the indices are different, these expressions are not like radicals. Option B, $\sqrt{5y}$, has a square root (index of 2), which matches our target expression. However, the radicand is $5y$, which is different from the radicand 5 in $3x\sqrt{5}$. Thus, these expressions are not like radicals either. Option C, $3(\sqrt[3]{5x})$, also has an index of 3, indicating a cube root. The radicand is $5x$, which differs from both the index and the radicand of our target expression. Therefore, option C is not a like radical. Finally, option D, $y\sqrt{5}$, has a square root (index of 2) and a radicand of 5, matching both the index and radicand of $3x\sqrt{5}$. The coefficient $y$ is different from $3x$, but this does not affect whether they are like radicals. Hence, $y\sqrt{5}$ is a like radical to $3x\sqrt{5}$.

This step-by-step analysis underscores the importance of carefully examining both the index and the radicand when identifying like radicals. A clear understanding of these components is essential for accurately determining which expressions can be combined and simplified. By breaking down each option and comparing it to the target expression, we can systematically eliminate the incorrect choices and arrive at the correct answer. The ability to perform this type of analysis is crucial for success in algebra and higher-level mathematics courses.

Detailed Analysis of the Options

To further solidify our understanding, let’s delve deeper into why each option is or is not a like radical to $3x\sqrt{5}$. This detailed examination will not only confirm our answer but also reinforce the principles behind identifying like radicals.

Option A: $x(\sqrt[3]{5})$

As mentioned earlier, option A, $x(\sqrt[3]{5})$, involves a cube root due to the index being 3. This immediately disqualifies it as a like radical to $3x\sqrt{5}$, which has a square root (index of 2). The index is a critical component in determining like radicals, and any difference in the index means the radicals cannot be combined. Furthermore, the expression inside the cube root is simply 5, which, while a constant, doesn’t change the fact that the root itself is different. Therefore, $x(\sqrt[3]{5})$ is not a like radical.

Option B: $\sqrt{5y}$

Option B, $\sqrt{5y}$, presents a square root, which matches the index of our target expression. However, the radicand is $5y$, which includes a variable $y$. This variable makes the radicand different from the radicand 5 in $3x\sqrt{5}$. Like radicals must have the same radicand; the presence of the variable $y$ means that $\sqrt{5y}$ cannot be combined with $3x\sqrt{5}$. The radicand must be identical for radicals to be considered “like.”

Option C: $3(\sqrt[3]{5x})$

Option C, $3(\sqrt[3]{5x})$, shares the same issue as option A: the index is 3, indicating a cube root. This difference in the index immediately rules it out as a like radical to $3x\sqrt{5}$. Additionally, the radicand is $5x$, which is different from 5. The combination of a different index and a different radicand makes it clear that this option is not a like radical. The presence of the variable $x$ under the cube root further emphasizes the dissimilarity between this expression and our target expression.

Option D: $y\sqrt{5}$

Option D, $y\sqrt{5}$, is the correct answer. It has a square root (index of 2), just like $3x\sqrt{5}$, and the radicand is 5, which matches our target expression. The coefficient $y$ is different from the coefficient $3x$, but as long as the index and radicand are the same, the expressions are like radicals. This means we could potentially combine these expressions if they were part of an addition or subtraction problem. The coefficient only affects the amount of the radical we have, not the type of radical itself.

By meticulously examining each option, we can confidently conclude that only option D, $y\sqrt{5}$, is a like radical to $3x\sqrt{5}$. This detailed analysis reinforces the importance of carefully considering both the index and the radicand when identifying like radicals, ensuring a solid understanding of this fundamental algebraic concept. Understanding these differences is crucial for successfully manipulating and simplifying radical expressions in various mathematical contexts.

Conclusion

In conclusion, when identifying like radicals of $3x\sqrt{5}$, the key is to look for expressions with the same index (2 in this case, indicating a square root) and the same radicand (5). Option D, $y\sqrt{5}$, is the only expression that meets both criteria. The coefficient $y$ is different from $3x$, but this does not affect their status as like radicals. Therefore, the correct answer is D. Understanding the concept of like radicals is crucial for simplifying complex algebraic expressions and solving equations involving radicals. Remember to always check both the index and the radicand when determining whether two radicals are like radicals. This skill is fundamental in algebra and provides a strong foundation for more advanced mathematical concepts. By mastering the identification of like radicals, students can confidently tackle more complex problems involving radical expressions, making it an essential tool in their mathematical toolkit. The ability to simplify and combine like radicals not only streamlines mathematical calculations but also enhances problem-solving efficiency and accuracy, which are highly valued in various fields, including engineering, physics, and computer science. Therefore, a thorough understanding of like radicals is indispensable for anyone pursuing a career in STEM or related disciplines. This comprehensive understanding enables students to approach mathematical challenges with greater confidence and precision, ultimately contributing to their academic and professional success.