Which Radical Is Like Cube Root Of 54 After Simplifying?

Determining like radicals involves simplifying the given radical expression and comparing it with the simplified forms of the options. In this article, we will delve into the process of simplifying radicals and identifying like radicals, using the example of $\sqrt[3]{54}$. We will explore each option, simplify them, and compare them to the simplified form of $\sqrt[3]{54}$ to find the correct answer. Understanding like radicals is crucial in simplifying radical expressions and performing operations such as addition and subtraction with radicals. Let’s embark on this mathematical journey to master the concept of like radicals.

Understanding Like Radicals

Before we dive into the problem, it's essential to understand what like radicals are. Like radicals, also known as similar radicals, are radical expressions that have the same index and the same radicand. The index is the small number outside the radical symbol (e.g., the '3' in $\sqrt[3]{54}$), and the radicand is the number inside the radical symbol (e.g., the '54' in $\sqrt[3]{54}$). For example, $2\sqrt{3}$ and $5\sqrt{3}$ are like radicals because they both have an index of 2 (since it's a square root) and a radicand of 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.

To identify like radicals, you often need to simplify the given radical expressions first. Simplifying radicals involves factoring the radicand and extracting any perfect powers that match the index. This process allows you to rewrite the radical in its simplest form, making it easier to compare with other radicals. Once simplified, you can easily determine if two radicals are like radicals by comparing their indices and radicands. Simplifying radicals is not just a matter of finding the smallest form; it is a fundamental step in determining relationships between different radical expressions.

The ability to identify like radicals is crucial for performing arithmetic operations on radical expressions. You can only add or subtract like radicals directly. For instance, $2\sqrt{3} + 5\sqrt{3}$ can be simplified to $7\sqrt{3}$, but $2\sqrt{3} + 5\sqrt{2}$ cannot be simplified further because the radicals are not alike. Understanding this principle helps in simplifying complex expressions and solving equations involving radicals. Simplifying radicals often involves prime factorization, which is a powerful tool for breaking down the radicand into its prime factors, making it easier to identify perfect powers. This step-by-step approach ensures that the radical is simplified to its most basic form, revealing whether it is similar to other radicals.

Simplifying $\sqrt[3]{54}$

The first step in solving the problem is to simplify the given radical, $\sqrt[3]{54}$. To simplify a cube root, we need to find the prime factorization of the radicand, which is 54 in this case. The prime factorization of 54 is $2 imes 3 imes 3 imes 3$, or $2 imes 3^3$. Now we can rewrite the radical as $\sqrt[3]{2 imes 3^3}$.

Using the properties of radicals, we can separate the factors inside the cube root: $\sqrt[3]{2 imes 3^3} = \sqrt[3]{3^3} imes \sqrt[3]{2}$. Since the cube root of $3^3$ is simply 3, we have $\sqrt[3]{3^3} imes \sqrt[3]{2} = 3\sqrt[3]{2}$. Thus, the simplified form of $\sqrt[3]{54}$ is $3\sqrt[3]{2}$. This simplified form is crucial because we will compare all other options to this to determine which one is a like radical.

Simplifying radicals often involves identifying perfect powers that match the index of the radical. In this case, we looked for perfect cubes within the factors of 54. This technique is essential for reducing radicals to their simplest form, which makes it easier to perform comparisons and operations. By understanding the properties of radicals, such as the product rule, we can effectively simplify expressions. The ability to factor and recognize perfect powers is a foundational skill in algebra and is particularly important when dealing with radicals. This systematic approach ensures that the radical is expressed in its most reduced form, which is necessary for identifying similar radicals.

Evaluating Option A: $\sqrt[3]{24}$

Now, let's evaluate option A, which is $\sqrt[3]{24}$. We need to simplify this radical and see if it is a like radical to $3\sqrt[3]{2}$. To simplify $\sqrt[3]{24}$, we first find the prime factorization of 24. The prime factorization of 24 is $2 imes 2 imes 2 imes 3$, or $2^3 imes 3$.

We can rewrite the radical as $\sqrt[3]{2^3 imes 3}$. Using the properties of radicals, we can separate the factors inside the cube root: $\sqrt[3]{2^3 imes 3} = \sqrt[3]{2^3} imes \sqrt[3]{3}$. The cube root of $2^3$ is 2, so we have $\sqrt[3]{2^3} imes \sqrt[3]{3} = 2\sqrt[3]{3}$.

Comparing $2\sqrt[3]{3}$ with the simplified form of the original radical, $3\sqrt[3]{2}$, we see that they have the same index (3) but different radicands (3 and 2, respectively). Therefore, $\sqrt[3]{24}$ is not a like radical to $\sqrt[3]{54}$. The process of simplifying radicals is essential for making accurate comparisons, and identifying the prime factors helps in finding perfect powers. This step-by-step evaluation ensures that each radical is correctly simplified and compared, leading to the correct identification of like radicals.

Evaluating Option B: $\sqrt[3]{162}$

Next, let's evaluate option B, which is $\sqrt[3]{162}$. We need to simplify this radical and compare it to $3\sqrt[3]{2}$. To simplify $\sqrt[3]{162}$, we start by finding the prime factorization of 162. The prime factorization of 162 is $2 imes 3 imes 3 imes 3 imes 3$, or $2 imes 3^4$.

We can rewrite the radical as $\sqrt[3]{2 imes 3^4}$. To simplify this, we look for perfect cubes within the factors. We can rewrite $3^4$ as $3^3 imes 3$. So, the radical becomes $\sqrt[3]{2 imes 3^3 imes 3}$. Now we can separate the factors: $\sqrt[3]{2 imes 3^3 imes 3} = \sqrt[3]{3^3} imes \sqrt[3]{2 imes 3}$.

The cube root of $3^3$ is 3, so we have $3\sqrt[3]{2 imes 3} = 3\sqrt[3]{6}$. Comparing $3\sqrt[3]{6}$ with the simplified form of the original radical, $3\sqrt[3]{2}$, we see that they have the same index (3) but different radicands (6 and 2, respectively). Therefore, $\sqrt[3]{162}$ is not a like radical to $\sqrt[3]{54}$. The key to simplifying radicals is to identify and extract perfect powers, ensuring that the remaining factors are in their simplest form. This methodical approach allows for a clear comparison of radicals.

Evaluating Option C: $\sqrt{128}$

Now, let's consider option C, which is $\sqrt{128}$. Notice that this is a square root, not a cube root like the original expression and the previous options. To be a like radical, the index must be the same. Since $\sqrt{128}$ is a square root (index 2) and our original radical $\sqrt[3]{54}$ is a cube root (index 3), they cannot be like radicals, regardless of the radicand. Thus, we can quickly conclude that option C is not a like radical.

To further illustrate this, we can simplify $\sqrt{128}$. The prime factorization of 128 is $2 imes 2 imes 2 imes 2 imes 2 imes 2 imes 2$, or $2^7$. We can rewrite the radical as $\sqrt{2^7}$. To simplify a square root, we look for perfect squares. We can rewrite $2^7$ as $2^6 imes 2$, which is $(2^3)^2 imes 2$. So, $\sqrt{2^7} = \sqrt{(2^3)^2 imes 2} = \sqrt{(8)^2 imes 2}$.

Separating the factors, we get $\sqrt{8^2} imes \sqrt{2} = 8\sqrt{2}$. Although we have simplified $\sqrt{128}$ to $8\sqrt{2}$, it is still a square root, while our original expression simplifies to a cube root. This difference in the index makes them unlike radicals. The recognition of the index as a crucial factor in determining like radicals is essential for avoiding unnecessary simplifications when the radicals are inherently different.

Evaluating Option D: $\sqrt[3]{128}$

Finally, let's evaluate option D, which is $\sqrt[3]{128}$. We need to simplify this cube root and compare it to $3\sqrt[3]{2}$. To simplify $\sqrt[3]{128}$, we find the prime factorization of 128. The prime factorization of 128 is $2 imes 2 imes 2 imes 2 imes 2 imes 2 imes 2$, or $2^7$.

We can rewrite the radical as $\sqrt[3]{2^7}$. To simplify a cube root, we look for perfect cubes within the factors. We can rewrite $2^7$ as $2^6 imes 2$, which is $(2^2)^3 imes 2$, or $4^3 imes 2$. So, the radical becomes $\sqrt[3]{4^3 imes 2}$. Now we can separate the factors: $\sqrt[3]{4^3 imes 2} = \sqrt[3]{4^3} imes \sqrt[3]{2}$.

The cube root of $4^3$ is 4, so we have $4\sqrt[3]{2}$. Comparing $4\sqrt[3]{2}$ with the simplified form of the original radical, $3\sqrt[3]{2}$, we see that they have the same index (3) and the same radicand (2). Therefore, $\sqrt[3]{128}$ is a like radical to $\sqrt[3]{54}$. The key to determining like radicals is the index and radicand, and this example clearly demonstrates that once simplified, the similarity is easily identifiable.

Conclusion

In conclusion, after simplifying the given options, we found that $\sqrt[3]{128}$ simplifies to $4\sqrt[3]{2}$, which is a like radical to the simplified form of $\sqrt[3]{54}$, which is $3\sqrt[3]{2}$. The other options, $\sqrt[3]{24}$ and $\sqrt[3]{162}$, had the same index but different radicands, and $\sqrt{128}$ had a different index altogether, making them unlike radicals.

Therefore, the correct answer is D. $\sqrt[3]{128}$.

Understanding how to simplify radicals and identify like radicals is crucial in algebra. This skill not only helps in solving problems involving radicals but also in performing more complex algebraic operations. By breaking down each radical into its simplest form, we can easily compare them and determine if they are like radicals. This step-by-step approach ensures accuracy and enhances understanding of radical expressions. Mastering these concepts builds a strong foundation for advanced mathematical topics.