Identifying Like Radicals Of 3x√5 A Comprehensive Guide

In the realm of mathematics, understanding the concept of like radicals is crucial for simplifying expressions and solving equations involving radicals. This comprehensive guide aims to delve into the intricacies of like radicals, specifically focusing on how to identify radicals that are similar to the expression $3x\sqrt{5}$. We'll explore the definition of like radicals, dissect the given expression, analyze the provided options, and ultimately determine the correct answer while providing a thorough explanation of the underlying principles. This exploration will not only help you solve this particular problem but also equip you with the knowledge to tackle similar problems with confidence. Whether you're a student grappling with radical expressions or simply someone looking to brush up on your math skills, this guide will serve as a valuable resource in your mathematical journey.

What are Like Radicals?

To effectively identify like radicals, it's essential to first define what they are. Like radicals, also known as similar radicals, are radical expressions that share the same index (the small number indicating the root, such as the '3' in a cube root) and the same radicand (the expression under the radical symbol). In simpler terms, they are radicals that differ only in their coefficients, which are the numbers multiplied by the radical expression. For instance, $2\sqrt{3}$, $-5\sqrt{3}$, and $\frac{1}{2}\sqrt{3}$ are all like radicals because they share the same index (2, implying a square root) and the same radicand (3). However, $\sqrt{2}$ and $\sqrt{3}$ are not like radicals because they have different radicands, even though they have the same index. Similarly, $\sqrt{5}$ and $\sqrt[3]{5}$ are not like radicals because they have different indices, despite sharing the same radicand.

Understanding this core concept is paramount for performing operations like addition and subtraction with radicals. Just as you can only combine like terms in algebraic expressions (e.g., $3x + 5x = 8x$), you can only combine like radicals. To add or subtract radicals, they must have the same index and radicand. The coefficients are then added or subtracted, while the radical part remains unchanged. For example, $4\sqrt{7} + 3\sqrt{7} = 7\sqrt{7}$. However, $2\sqrt{5} + 3\sqrt{2}$ cannot be simplified further because the radicals are not alike. The ability to identify and manipulate like radicals is therefore a fundamental skill in algebra and is crucial for simplifying complex expressions and solving equations involving roots.

Dissecting the Given Radical: $3x\sqrt{5}$

Now, let's carefully examine the given radical expression: $3x\sqrt{5}$. This expression is composed of several parts that are crucial to understanding its nature and identifying like radicals. First, we have the coefficient, which is $3x$. This part represents the number multiplying the radical. It's important to note that the coefficient can be a constant, a variable, or a combination of both. In this case, it's a combination of a constant (3) and a variable (x). Next, we have the radical part, which consists of the radical symbol ($\sqrt{ }$) and the radicand. The radicand is the expression under the radical symbol, and in this case, it is 5. Finally, the index, although not explicitly written, is implicitly 2, indicating a square root. Remember, the square root symbol ($\sqrt{ }$) is conventionally used when the index is 2. If the index were any other number, it would be written as a superscript before the radical symbol, such as $\sqrt[3]{ }$ for a cube root or $\sqrt[4]{ }$ for a fourth root.

To find a like radical to $3x\sqrt{5}$, we need to focus on the radical part, specifically the index and the radicand. A like radical must have the same index (2) and the same radicand (5). The coefficient can be different, and it can even involve variables, as long as the radical part remains the same. This understanding is key to correctly evaluating the given options. By breaking down the given expression into its components – coefficient, index, and radicand – we gain a clear picture of what constitutes a like radical. This systematic approach allows us to analyze each option methodically and eliminate those that do not meet the criteria. In the subsequent sections, we will apply this understanding to the provided options and determine which one is indeed a like radical to the given expression.

Analyzing the Options: A Step-by-Step Evaluation

To determine which of the given options is a like radical to $3x\sqrt{5}$, we must meticulously examine each option and compare its radical part (index and radicand) with that of the given expression. Remember, for a radical to be considered "like," it must have the same index and the same radicand as $3x\sqrt{5}$, which has an index of 2 and a radicand of 5.

Let's analyze each option:

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

In this option, the radical part is $\sqrt[3]{5}$. The index is 3 (cube root), and the radicand is 5. While the radicand matches the given expression, the index does not. The given expression has an index of 2 (square root), whereas this option has an index of 3. Therefore, option A is not a like radical.

B. $\sqrt{5y}$

Here, the radical part is $\sqrt{5y}$. The index is 2 (square root), which matches the given expression. However, the radicand is $5y$, which is different from the radicand of the given expression, which is simply 5. The presence of the variable 'y' under the radical makes this option unlike the given expression. Thus, option B is not a like radical.

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

In this case, the radical part is $\sqrt[3]{5x}$. The index is 3 (cube root), which does not match the index of 2 in the given expression. Furthermore, the radicand is $5x$, which is also different from the radicand of 5 in the given expression. Therefore, option C is not a like radical.

D. $y\sqrt{5}$

Finally, we have option D, where the radical part is $\sqrt{5}$. The index is 2 (square root), which matches the given expression, and the radicand is 5, which also matches. The coefficient is 'y', which, as we know, can be different from the coefficient in the given expression ($3x$) without affecting whether the radicals are "like." Therefore, option D is a like radical.

By systematically evaluating each option based on the criteria for like radicals, we can confidently identify the correct answer.

The Solution: Identifying the Like Radical

After a thorough analysis of each option, it becomes clear that option D, $y\sqrt{5}$, is the like radical to $3x\sqrt{5}$. This conclusion is based on the fundamental definition of like radicals, which states that they must have the same index and the same radicand. Let's reiterate why option D meets these criteria:

• Index: Both $3x\sqrt{5}$ and $y\sqrt{5}$ have an index of 2, implying a square root. The absence of an explicitly written index on the radical symbol signifies a square root.
• Radicand: Both expressions have the same radicand, which is 5. The radicand is the expression under the radical symbol, and in both cases, it is simply the number 5.
• Coefficient: While the coefficients are different ($3x$ in the given expression and $y$ in option D), the coefficients do not determine whether radicals are like. Like radicals can have different coefficients, and it is the similarity in the index and radicand that defines them.

Options A, B, and C were ruled out because they failed to meet at least one of the criteria for like radicals:

• Option A, $x(\sqrt[3]{5})$, has a different index (3) compared to the given expression (2).
• Option B, $\sqrt{5y}$, has a different radicand ($5y$) compared to the given expression (5).
• Option C, $3(\sqrt[3]{5x})$, has both a different index (3) and a different radicand ($5x$) compared to the given expression.

Therefore, the correct answer is unequivocally D. $y\sqrt{5}$. Understanding the nuances of like radicals is essential for simplifying radical expressions and performing operations like addition and subtraction. By focusing on the index and radicand, one can accurately identify like radicals and manipulate them effectively in various mathematical contexts.

Key Takeaways and Further Exploration

This exploration of like radicals, centered around the expression $3x\sqrt{5}$, provides several key takeaways that are crucial for mastering radical expressions and related mathematical concepts. Firstly, the definition of like radicals is paramount. Like radicals must possess the same index and the same radicand, irrespective of their coefficients. This understanding forms the foundation for identifying and manipulating like radicals effectively. Secondly, the process of dissecting a radical expression into its components – coefficient, index, and radicand – is a valuable technique for analyzing and comparing radical expressions. By breaking down expressions into their constituent parts, we can systematically evaluate whether they meet the criteria for being like radicals.

Thirdly, the ability to analyze options methodically and eliminate those that do not fit the definition is an important problem-solving skill in mathematics. By carefully comparing the index and radicand of each option with the given expression, we can confidently arrive at the correct answer. In this case, we saw how options A, B, and C failed to meet the criteria for like radicals, leaving option D as the only viable choice.

To further enhance your understanding of radicals, consider exploring related concepts such as simplifying radicals, rationalizing the denominator, and performing arithmetic operations with radicals. Simplifying radicals involves expressing them in their simplest form by factoring out perfect squares (or perfect cubes, etc.) from the radicand. Rationalizing the denominator involves eliminating radicals from the denominator of a fraction, which is often necessary for simplifying expressions and comparing them effectively. Mastering these concepts will provide a more comprehensive understanding of radicals and their applications in various areas of mathematics. By practicing with a variety of examples and seeking out additional resources, you can solidify your knowledge and develop the skills necessary to confidently tackle more complex problems involving radicals.