Find The Like Radical Of $3x\sqrt{5}$ From The Following Options. A. $x\sqrt[3]{5}$ B. $\sqrt{5y}$ C. $3\sqrt[3]{5x}$ D. $y\sqrt{5}$
Introduction to Like Radicals
In mathematics, particularly when dealing with radicals, the concept of like radicals is fundamental for simplifying expressions and performing operations such as addition and subtraction. To effectively grasp this concept, it's essential to understand what radicals are and the conditions that must be met for them to be considered "like." In the context of algebraic expressions, radicals involve roots, such as square roots, cube roots, and higher-order roots. The anatomy of a radical expression includes the radical symbol (√), the radicand (the expression under the radical symbol), and the index (the degree of the root). For instance, in the expression $\sqrt[n]{a}$, 'n' represents the index, and 'a' is the radicand. When the index is 2, it denotes a square root, and the index is often omitted for brevity, as in $\sqrt{a}$. When dealing with the concept of like radicals, it’s essential to understand what radicals are and the conditions that must be met for them to be considered “like.” This involves looking at the index, the radicand, and how these components interact within algebraic expressions. The index determines the type of root being taken (square root, cube root, etc.), while the radicand is the value under the radical sign. The interplay between these components is crucial in determining whether radicals can be combined or simplified together. The ability to identify and work with like radicals is not only crucial for simplifying mathematical expressions but also for solving equations and understanding more advanced algebraic concepts. Students who master this skill will find it easier to tackle complex problems and gain a deeper appreciation for the elegance and structure of mathematics. Therefore, a solid understanding of like radicals forms a cornerstone of algebraic proficiency.
Defining Like Radicals: Index and Radicand
Like radicals are defined as radicals that share the same index and the same radicand. This definition is crucial because only like radicals can be combined through addition and subtraction. The index, as mentioned earlier, specifies the type of root being taken, while the radicand is the expression under the radical symbol. For example, $2\sqrt{3}$ and $5\sqrt{3}$ are like radicals because both have an index of 2 (square root) and a radicand of 3. In contrast, $\sqrt{2}$ and $\sqrt{3}$ are not like radicals because, although they share the same index, their radicands are different. Similarly, $\sqrt{5}$ and $\sqrt[3]{5}$ are not like radicals because they have different indices. When assessing whether radicals are alike, it's important to focus on both the index and the radicand, as a mismatch in either will disqualify them from being considered like radicals. Understanding this definition is the first step in simplifying expressions and solving equations involving radicals. This concept extends beyond simple numerical examples and applies to algebraic expressions as well. Consider the expressions $x\sqrt{y}$ and $z\sqrt{y}$. These are like radicals as long as 'y' remains the same under the square root, regardless of the coefficients 'x' and 'z'. This principle allows us to combine such terms, similar to how we combine like terms in polynomials. The coefficient outside the radical does not affect whether radicals are "like"; it only plays a role when combining them. Therefore, accurately identifying like radicals is a critical skill in algebra, paving the way for more complex operations and problem-solving strategies. Students who master this concept can confidently navigate radical expressions and equations, leading to a deeper understanding of mathematical principles.
Analyzing the Given Radical: $3x\sqrt{5}$
To determine which of the provided options is a like radical to $3x\sqrt5}$, we must first dissect the given expression. The expression $3x\sqrt{5}$ consists of a coefficient $3x$ multiplied by the square root of 5. Here, the index of the radical is 2 (since it is a square root), and the radicand is 5. Recall that for a radical to be considered “like” to $3x\sqrt{5}$, it must also have an index of 2 and a radicand of 5. The coefficient, in this case $3x$, does not influence whether the radicals are “like”; it only affects the magnitude of the term. Therefore, any radical with $\sqrt{5}$ as its radical part will be a like radical to $3x\sqrt{5}$. This understanding helps narrow down the options and focus on the essential components of the radical expression. The presence of a variable outside the radical, such as the 'x' in $3x\sqrt{5}$, might initially cause confusion, but it's crucial to remember that like radicals are determined solely by the index and the radicand. The variable coefficient simply acts as a multiplier for the entire radical term. When we examine the structure of $3x\sqrt{5}$, we can see that it is a product of three components$. To find a like radical, we need to match the radical part, which is $\sqrt{5}$. This analysis sets the stage for evaluating each of the provided options against these criteria, ensuring that we select the correct like radical based on the fundamental definition. In summary, to identify a like radical to $3x\sqrt{5}$, we are looking for an expression that also involves the square root of 5, irrespective of any coefficients or variables outside the radical.
Evaluating Option A: $x\sqrt[3]{5}$
Option A presents the expression $x\sqrt[3]{5}$. To determine if this is a like radical to $3x\sqrt{5}$, we need to compare its index and radicand with those of the given expression. The index of $x\sqrt[3]{5}$ is 3, as indicated by the cube root symbol. The radicand is 5, which is the same as the radicand in $3x\sqrt{5}$. However, the key difference lies in the index. The given expression, $3x\sqrt{5}$, has an index of 2 (square root), while $x\sqrt[3]{5}$ has an index of 3 (cube root). Since like radicals must have the same index and the same radicand, and in this case, the indices are different, Option A is not a like radical to $3x\sqrt{5}$. This distinction is critical in understanding the conditions necessary for radicals to be combined or simplified together. The presence of the variable 'x' outside the radical is a distraction in this comparison, as it does not influence whether the radicals are alike. The focus remains solely on the index and the radicand. It’s also important to recognize that cube roots and square roots represent fundamentally different operations. A square root seeks a number that, when multiplied by itself, yields the radicand, while a cube root seeks a number that, when multiplied by itself twice, yields the radicand. This difference in operation further underscores why radicals with different indices cannot be considered “like.” Therefore, the presence of a cube root in $x\sqrt[3]{5}$ immediately disqualifies it as a like radical to $3x\sqrt{5}$, which involves a square root.
Examining Option B: $\sqrt{5y}$
Now, let's consider Option B, which presents the expression $\sqrt{5y}$. To assess whether this is a like radical to $3x\sqrt{5}$, we must again compare the index and the radicand. The index of $\sqrt{5y}$ is 2, as it is a square root. However, the radicand is $5y$, which is different from the radicand of the given expression, which is simply 5. For radicals to be considered “like,” they must have both the same index and the same radicand. In this case, while the index matches, the radicand does not, because of the presence of the variable 'y' inside the radical. The expression $\sqrt{5y}$ represents the square root of the product of 5 and 'y', whereas $3x\sqrt{5}$ represents 3x multiplied by the square root of 5. This difference in the radicand means that $\sqrt{5y}$ cannot be combined directly with $3x\sqrt{5}$ through addition or subtraction, which is a key characteristic of like radicals. The presence of the variable 'y' under the radical sign changes the fundamental nature of the expression, preventing it from being a like radical to $3x\sqrt{5}$. It’s important to understand that the radicand must be identical for radicals to be considered like, and this includes not only the numerical value but also any variables and their exponents. In this scenario, the 'y' within the radical alters the expression, making it distinct from $3x\sqrt{5}$. Therefore, Option B, $\sqrt{5y}$, is not a like radical to the given expression because their radicands differ. This analysis highlights the importance of carefully examining the radicand when identifying like radicals.
Analyzing Option C: $3\sqrt[3]{5x}$
Option C presents the expression $3\sqrt[3]{5x}$. To determine if it's a like radical to $3x\sqrt{5}$, we need to analyze its index and radicand. The index of $3\sqrt[3]{5x}$ is 3, which indicates a cube root. The radicand is $5x$, which is the expression under the cube root symbol. Comparing this to the given expression, $3x\sqrt{5}$, we immediately notice a difference in the index. The given expression has an index of 2 (square root), while Option C has an index of 3 (cube root). Additionally, the radicands are different; the given expression has a radicand of 5, whereas Option C has a radicand of $5x$. Since like radicals must have both the same index and the same radicand, Option C does not meet these criteria. The difference in the index alone is sufficient to disqualify $3\sqrt[3]{5x}$ as a like radical to $3x\sqrt{5}$. The presence of the variable 'x' within the radicand further emphasizes this distinction. The expression $3\sqrt[3]{5x}$ represents 3 times the cube root of 5x, which is a fundamentally different operation and expression compared to $3x\sqrt{5}$. It is crucial to recognize that radicals with different indices cannot be combined or simplified together, and the same applies when the radicands are not identical. Therefore, Option C, $3\sqrt[3]{5x}$, is not a like radical to the given expression due to differences in both the index and the radicand. This analysis reinforces the importance of a thorough comparison of both the index and the radicand when identifying like radicals.
Identifying the Like Radical: Option D, $y\sqrt{5}$
Finally, let's examine Option D, which gives us the expression $y\sqrt{5}$. When we compare this to the original expression, $3x\sqrt{5}$, we focus on the index and the radicand. The index of $y\sqrt{5}$ is 2, as it is a square root, and the radicand is 5. This matches the index and radicand of the given expression $3x\sqrt{5}$. The coefficients outside the radical, $3x$ in the original expression and 'y' in Option D, do not affect whether the radicals are “like”. The crucial factor is that both expressions involve the square root of 5. Therefore, $y\sqrt{5}$ is a like radical to $3x\sqrt{5}$. This means that these two terms could potentially be combined if they appeared in the same expression, such as in the simplification of algebraic equations. The variable 'y' simply acts as a coefficient, similar to how $3x$ acts as a coefficient in the original expression. Understanding this distinction is key to correctly identifying like radicals and simplifying expressions involving them. Option D satisfies the essential requirement of having the same index and radicand as the original expression, making it the correct answer. This highlights the importance of focusing on the radical part of the expression when determining whether radicals are “like”. In summary, $y\sqrt{5}$ is a like radical to $3x\sqrt{5}$ because they both involve the square root of 5, irrespective of the coefficients outside the radical.
Conclusion: $y\sqrt{5}$ is the Like Radical
In conclusion, after analyzing all the options, we have determined that Option D, $y\sqrt{5}$, is the like radical to $3x\sqrt{5}$. This determination was made by comparing the index and radicand of each option with those of the given expression. Like radicals must have the same index and the same radicand, and only Option D satisfies both of these conditions. Options A and C were ruled out because they involved a cube root (index of 3), while the given expression involves a square root (index of 2). Option B was eliminated because its radicand was $5y$, which is different from the radicand of 5 in the given expression. Option D, $y\sqrt{5}$, has an index of 2 (square root) and a radicand of 5, making it a like radical to $3x\sqrt{5}$. The coefficients outside the radical do not affect whether radicals are alike, so the presence of 'y' in $y\sqrt{5}$ and $3x$ in $3x\sqrt{5}$ does not change the fact that they are like radicals. This exercise underscores the importance of a clear understanding of the definition of like radicals and the ability to identify the index and radicand in radical expressions. Mastering this concept is crucial for simplifying expressions, solving equations, and further studies in algebra and mathematics. The ability to accurately identify like radicals is a fundamental skill that enables students to confidently manipulate and simplify complex algebraic expressions, laying the groundwork for more advanced mathematical concepts.