Like Radicals: Find The Match For $\sqrt[3]{7x}$

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Hey guys! Let's dive into the fascinating world of radicals, specifically focusing on identifying like radicals. If you've ever felt a bit lost in the maze of square roots, cube roots, and beyond, you're in the right place. Today, we're tackling the question: Which of the following is a like radical to $\sqrt[3]{7x}$? We'll break down the concept of like radicals, walk through the solution step-by-step, and equip you with the knowledge to confidently handle similar problems. So, grab your metaphorical math helmets, and let's get started!

What are Like Radicals?

Before we jump into the specific question, let's make sure we're all on the same page about what like radicals actually are. In simple terms, like radicals are radicals that have the same index (the little number indicating the root, like the '3' in a cube root) and the same radicand (the expression under the radical sign). Think of it like this: like radicals are the identical twins of the radical world. They might have different coefficients (the numbers in front of the radical), but their core structure is the same.

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. However, $\sqrt{5}$ and $\sqrt{7}$ are not like radicals because they have different radicands, even though their index is the same. Similarly, $\sqrt{5}$ and $\sqrt[3]{5}$ are not like radicals because they have different indices, even though their radicand is the same.

Understanding this fundamental concept is crucial for simplifying expressions, performing operations with radicals, and, of course, answering questions like the one we're tackling today. So, with this definition in mind, let's revisit our original radical, $\sqrt[3]{7x}$, and see what other radicals share its unique DNA.

Analyzing the Given Options

Now that we know what like radicals are, let's analyze the options presented in our question. Remember, we're looking for a radical that has both the same index (3, indicating a cube root) and the same radicand (7x) as $\sqrt[3]{7x}$. Let's break down each option:

A. $4(\sqrt[3]{7x})$

At first glance, this option looks promising. We see the cube root symbol and the expression 7x. But let's take a closer look. The radical here is $\sqrt[3]{7x}$. It has an index of 3 and a radicand of 7x. The '4' in front is simply a coefficient, and coefficients don't affect whether radicals are "like" or not. This is like having 4 apples – it's still an apple, just more of them! So, this option appears to be a strong contender.

B. $\sqrt{7x}$

This option might seem similar at first, but there's a crucial difference. Notice that there's no little number indicating the index. When there's no index written, it's understood to be 2, representing a square root. So, this is actually $\sqrt[2]{7x}$. While the radicand (7x) is the same as our original radical, the index is different (2 instead of 3). Remember, like radicals need to have both the same index and the same radicand, so this option is not a match.

C. $x(\sqrt[3]{7})$

This option also has a cube root (index of 3), which is a good start. However, let's examine the radicand. Here, the radicand is just '7', not '7x'. The 'x' is outside the radical, acting as a coefficient. Since the radicand is different from our original radical's radicand (7x), this option is not a like radical.

D. $7\sqrt{x}$

This option presents a square root (index of 2) with a radicand of 'x'. Both the index and the radicand are different from our original radical $\sqrt[3]{7x}$ (index of 3, radicand of 7x). Therefore, this option is not a like radical.

The Verdict: Identifying the Like Radical

After carefully analyzing each option, we've determined that only one option perfectly matches the criteria for a like radical: it has the same index (3) and the same radicand (7x) as our original radical, $\sqrt[3]{7x}$.

The correct answer is A. $4(\sqrt[3]{7x})$.

This radical is a like radical to $\sqrt[3]{7x}$ because it shares the same index (3) and the same radicand (7x). The coefficient '4' simply indicates that we have four times the quantity of the cube root of 7x, but it doesn't change the fundamental nature of the radical itself.

Why This Matters: The Importance of Like Radicals

Now that we've successfully identified the like radical, you might be wondering, "Why does this even matter?" Well, understanding like radicals is crucial for several key operations in algebra and beyond. The most important reason is that you can only add or subtract like radicals. Just like you can only combine "apples" with "apples" and "oranges" with "oranges," you can only combine like radicals.

For instance, you can simplify the expression $2\sqrt{5} + 3\sqrt{5}$ because both terms have the same radical part ($\sqrt{5}$). You can simply add the coefficients: 2 + 3 = 5, resulting in $5\sqrt{5}$. However, you cannot directly simplify $2\sqrt{5} + 3\sqrt{7}$ because the radicals are not alike. You'd need to explore other techniques, like simplifying the radicals further if possible, before attempting to combine them.

This principle extends to more complex expressions and equations involving radicals. When solving equations, simplifying expressions, or performing other algebraic manipulations, identifying and combining like radicals is a fundamental step. Without this understanding, you'd be stuck trying to add "apples" and "oranges," leading to incorrect results.

Furthermore, the concept of like radicals is essential when dealing with operations like rationalizing denominators and simplifying radical expressions in general. These skills are not only important for algebra but also lay the groundwork for more advanced topics in mathematics, such as calculus and trigonometry.

So, mastering the concept of like radicals is not just about answering specific questions; it's about building a solid foundation for your mathematical journey. It's a key that unlocks a wide range of problem-solving techniques and empowers you to tackle more complex challenges with confidence.

Practice Makes Perfect: Sharpening Your Radical Skills

Like any mathematical concept, mastering like radicals requires practice. The more you work with them, the more comfortable and confident you'll become. So, let's take a look at some additional practice problems to help solidify your understanding.

Practice Problem 1: Which of the following is a like radical to $-3\sqrt{2x}$?

A. $5\sqrt{2x}$ B. $-3\sqrt{3x}$ C. $2\sqrt[3]{2x}$ D. $4x\sqrt{2}$

Solution:

Let's break this down. We're looking for a radical with the same index (2, since it's a square root) and the same radicand (2x) as $-3\sqrt{2x}$.

A. $5\sqrt{2x}$ has the same index and radicand, so it's a like radical. B. $-3\sqrt{3x}$ has the same index but a different radicand (3x), so it's not a like radical. C. $2\sqrt[3]{2x}$ has a different index (3), so it's not a like radical. D. $4x\sqrt{2}$ has the same index but a different radicand (2), so it's not a like radical.

Therefore, the correct answer is A. $5\sqrt{2x}$.

Practice Problem 2: Simplify the following expression: $4\sqrt[3]{5} - 2\sqrt[3]{5} + \sqrt[3]{5}$

Solution:

Notice that all three terms are like radicals – they all have an index of 3 and a radicand of 5. So, we can combine them by adding and subtracting their coefficients:

4 - 2 + 1 = 3

Therefore, the simplified expression is $3\sqrt[3]{5}$.

Practice Problem 3: Which of the following radicals cannot be combined with $7\sqrt{11}$?

A. $-2\sqrt{11}$ B. $\sqrt{11}$ C. $3\sqrt{22}$ D. $5\sqrt{11} + \sqrt{11}$

Solution:

We're looking for a radical that is not a like radical to $7\sqrt{11}$. This means it should have either a different index or a different radicand.

A. $-2\sqrt{11}$ is a like radical. B. $\sqrt{11}$ is a like radical. C. $3\sqrt{22}$ has a different radicand (22), so it's not a like radical. D. $5\sqrt{11} + \sqrt{11}$ can be simplified to $6\sqrt{11}$, which is a like radical.

Therefore, the correct answer is C. $3\sqrt{22}$.

By working through these practice problems, you're actively reinforcing your understanding of like radicals and honing your skills in identifying them. Remember, the key is to focus on the index and the radicand – if they match, you've found a like radical!

Wrapping Up: Your Journey to Radical Mastery

So, there you have it! We've successfully navigated the world of like radicals, dissected the question of which radical is like $\sqrt[3]{7x}$, and uncovered the importance of this concept in simplifying expressions and solving equations. Remember, like radicals are the identical twins of the radical world – they share the same index and the same radicand, allowing us to combine them through addition and subtraction.

By understanding this fundamental principle, you've not only answered a specific question but also gained a valuable tool for your mathematical toolkit. You're now equipped to tackle more complex problems involving radicals with confidence and precision.

Keep practicing, keep exploring, and keep challenging yourself. The world of mathematics is full of fascinating concepts waiting to be discovered, and you're well on your way to becoming a radical master! Keep up the great work, guys!