Identifying Like Radicals A Guide To Similar Radical Expressions

When delving into the realm of radicals in mathematics, a fundamental concept to grasp is that of "like radicals." Like radicals, in essence, 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 sign). Identifying like radicals is crucial for simplifying expressions, performing operations like addition and subtraction, and solving radical equations. This article aims to provide a comprehensive understanding of like radicals, focusing on how to identify them and applying this knowledge to the specific question: Which of the following is a like radical to ${\sqrt[3]{7x}}$? We will dissect the definition of like radicals, explore various examples, and meticulously analyze each option presented in the question to arrive at the correct answer. Understanding like radicals is not just a mathematical exercise; it's a foundational skill that unlocks more advanced concepts in algebra and beyond.

The importance of recognizing like radicals stems from their role in simplifying and manipulating radical expressions. Just as we can only combine like terms (e.g., 3x + 2x = 5x), we can only add or subtract like radicals. This ability to combine like radicals simplifies complex expressions into more manageable forms, making them easier to work with in further calculations or problem-solving scenarios. Furthermore, the concept of like radicals extends to various mathematical applications, including solving equations, simplifying expressions in calculus, and dealing with complex numbers. Therefore, a solid understanding of this concept is not just beneficial but essential for anyone pursuing mathematics or related fields.

Before we delve into the specific question, let's solidify our understanding of what constitutes a like radical. A radical expression consists of three main components: the index, the radicand, and the coefficient (the number multiplying the radical). For two or more radical expressions to be considered "like," they must have the same index and the same radicand. The coefficient, however, can be different. For example, 2${\sqrt{5}}$ and -7${\sqrt{5}}$ are like radicals because they both have an index of 2 (since it's a square root) and a radicand of 5, even though their coefficients are different. On the other hand, ${\sqrt{5}}$ and ${\sqrt[3]{5}}$ are not like radicals because they have different indices, and ${\sqrt{5}}$ and ${\sqrt{7}}$ are not like radicals because they have the same index but different radicands. This fundamental distinction is key to accurately identifying like radicals in various expressions and forms the basis for our analysis of the given question.

Dissecting the Question: Identifying Like Radicals

To effectively answer the question, "Which of the following is a like radical to ${\sqrt[3]{7x}}$?", we must first meticulously analyze the given expression, ${\sqrt[3]{7x}}$, and then compare it against each of the provided options. The given expression is a cube root (index of 3) with a radicand of 7x. Therefore, any like radical must also have an index of 3 and a radicand of 7x, irrespective of the coefficient multiplying the radical. This understanding forms the basis for our systematic evaluation of each option.

Option 1: ${7\sqrt{x}}$

Let's begin by examining the first option, ${7\sqrt{x}}$. This expression represents 7 times the square root of x. Notice that the index here is 2 (since it's a square root), and the radicand is x. Comparing this to our original expression, ${\sqrt[3]{7x}}$, we see a clear difference in both the index and the radicand. The original expression has an index of 3 and a radicand of 7x, while this option has an index of 2 and a radicand of x. Therefore, ${7\sqrt{x}}$ is not a like radical to ${\sqrt[3]{7x}}$. This highlights the importance of paying close attention to both the index and the radicand when identifying like radicals. Even if the radicands shared a common variable, the differing indices immediately disqualify them as like radicals.

Option 2: ${4(\sqrt[3]{7x})}$

Next, we consider the second option: ${4(\sqrt[3]{7x})}$. This expression represents 4 times the cube root of 7x. A careful comparison with our original expression, ${\sqrt[3]{7x}}$, reveals a striking similarity. Both expressions have an index of 3 and a radicand of 7x. The only difference is the coefficient: the original expression can be thought of as having a coefficient of 1, while this option has a coefficient of 4. As we established earlier, like radicals can have different coefficients. Therefore, ${4(\sqrt[3]{7x})}$ is a like radical to ${\sqrt[3]{7x}}$. This option perfectly illustrates the definition of like radicals, emphasizing that the index and radicand must match, while the coefficient can vary.

Option 3: ${\sqrt{7x}}$

Now, let's analyze the third option, ${\sqrt{7x}}$. This expression represents the square root of 7x. Here, the index is 2 (since it's a square root), and the radicand is 7x. Comparing this to our original expression, ${\sqrt[3]{7x}}$, we observe that the radicands are the same (7x), but the indices are different. The original expression has an index of 3, while this option has an index of 2. Since like radicals must have the same index and the same radicand, ${\sqrt{7x}}$ is not a like radical to ${\sqrt[3]{7x}}$. This example reinforces the critical role of the index in determining whether radicals are "like." Even with identical radicands, differing indices preclude radicals from being considered like terms.

Option 4: ${x(\sqrt[3]{7})}$

Finally, we evaluate the fourth option, ${x(\sqrt[3]{7})}$. This expression represents x times the cube root of 7. In this case, the index is 3, and the radicand is 7. When we compare this to our original expression, ${\sqrt[3]{7x}}$, we notice that while the indices match (both are cube roots), the radicands are different. The original expression has a radicand of 7x, while this option has a radicand of 7. Since like radicals must have the same index and the same radicand, ${x(\sqrt[3]{7})}$ is not a like radical to ${\sqrt[3]{7x}}$. This option highlights the importance of ensuring that both the index and the radicand are identical for radicals to be considered "like." Even a subtle difference in the radicand, such as the presence of the variable 'x', can disqualify two radicals from being like terms.

Conclusion: The Correct Answer and Its Significance

After a thorough examination of each option, we can definitively conclude that the correct answer is Option 2: ${4(\sqrt[3]{7x})}$. This is the only expression among the choices that shares both the same index (3) and the same radicand (7x) as the original expression, ${\sqrt[3]{7x}}$. The coefficient of 4 does not affect its status as a like radical, as like radicals can have different coefficients.

The importance of correctly identifying like radicals extends beyond this specific question. It's a foundational skill for simplifying radical expressions, performing operations with radicals, and solving radical equations. When simplifying expressions, we can only combine like radicals through addition or subtraction, similar to how we combine like terms in algebraic expressions. For example, if we had the expression ${3\sqrt[3]{7x} + 4\sqrt[3]{7x}}$, we could simplify it to ${7\sqrt[3]{7x}}$ because the radicals are alike. However, we could not combine ${3\sqrt[3]{7x} + \sqrt{7x}}$ because the radicals are not alike due to the different indices.

In summary, understanding the concept of like radicals is crucial for success in algebra and beyond. By carefully analyzing the index and radicand of radical expressions, we can accurately identify like radicals and perform necessary simplifications and operations. This skill is not just about finding the right answer to a specific question; it's about building a solid foundation for more advanced mathematical concepts. The ability to discern like radicals is a fundamental tool in the mathematician's toolkit, enabling them to navigate the complexities of radical expressions and equations with confidence and precision.