Simplifying Radicals $21 \sqrt[3]{15}-9 \sqrt[3]{15}$ A Step By Step Guide

In mathematics, simplifying expressions involving radicals is a fundamental skill. This article focuses on how to simplify expressions with radicals, specifically addressing the expression $21 \sqrt[3]{15}-9 \sqrt[3]{15}$. We will walk through the steps to correctly simplify this expression and explain the reasoning behind each step. This guide is designed to help students and anyone interested in mathematics to better understand how to handle radical expressions. You'll learn how to identify like terms, combine them, and arrive at the simplest form of the expression. The ability to simplify radical expressions is crucial for more advanced mathematical topics, making this a valuable skill to master. Grasping the concepts presented here will provide a solid foundation for tackling more complex problems involving radicals. Let’s dive in and explore the process together!

Understanding Radicals

Before diving into the specific problem, it’s crucial to understand what radicals are and how they work. A radical is a mathematical expression that involves a root, such as a square root, cube root, or any higher-order root. The general form of a radical is $\sqrt[n]{a}$, where n is the index (the degree of the root) and a is the radicand (the number under the radical sign). For instance, in the expression $\sqrt[3]{15}$, the index is 3, indicating a cube root, and the radicand is 15. When simplifying radicals, the key is to identify perfect powers within the radicand. A perfect power is a number that can be expressed as an integer raised to an integer power. For example, 8 is a perfect cube because $8 = 2^3$. Recognizing perfect powers allows us to simplify radicals by extracting the root. Understanding these basics is crucial because simplifying radical expressions often involves combining like terms, which means identifying terms that have the same index and radicand. Without a firm grasp of these fundamental concepts, simplifying expressions like $21 \sqrt[3]{15}-9 \sqrt[3]{15}$ can be challenging. Therefore, ensuring you understand the core principles of radicals will set you up for success in tackling more complex problems. Let's move on to how we can apply these principles to solve the given problem.

Identifying Like Terms

To effectively simplify the expression $21 \sqrt[3]{15}-9 \sqrt[3]{15}$, the first critical step is to identify like terms. Like terms, in the context of radical expressions, are terms that have the same radicand and the same index. In other words, they must have the same number under the radical sign and the same type of root (square root, cube root, etc.). In the given expression, we have two terms: $21 \sqrt[3]{15}$ and $-9 \sqrt[3]{15}$. Both terms have the same radicand, which is 15, and the same index, which is 3 (indicating a cube root). This means that they are indeed like terms. Identifying like terms is crucial because you can only combine terms that are alike. Trying to combine terms with different radicands or indices is like trying to add apples and oranges—it simply cannot be done directly. Once you've identified the like terms, you can proceed with the next step, which involves combining these terms by performing the necessary arithmetic operations on their coefficients. The coefficients are the numbers in front of the radical expression, in this case, 21 and -9. Recognizing and grouping like terms is a fundamental skill in simplifying radical expressions, and mastering this step is key to solving more complex problems in algebra and beyond. Let’s proceed to the next phase, where we combine these like terms to simplify the expression.

Combining Like Terms

Having identified $21 \sqrt[3]{15}$ and $-9 \sqrt[3]{15}$ as like terms, the next step is to combine them. Combining like terms in radical expressions is similar to combining like terms in algebraic expressions. You simply perform the arithmetic operation on the coefficients of the like terms while keeping the radical part the same. In this case, we need to subtract the coefficients: $21 - 9$. Performing this subtraction, we get $21 - 9 = 12$. Now, we take this result and multiply it by the radical part, which is $\sqrt[3]{15}$. So, the simplified expression becomes $12 \sqrt[3]{15}$. This process is analogous to combining terms like $21x - 9x$, where you would subtract the coefficients (21 and 9) and keep the variable x. The radical $\sqrt[3]{15}$ behaves just like the variable x in this context. It’s important to remember that you can only combine terms that are truly alike—same index and same radicand. This step of combining like terms is a crucial part of simplifying radical expressions and is a building block for more complex algebraic manipulations. By accurately combining like terms, we reduce the expression to its simplest form, making it easier to understand and work with in further calculations. Let’s take a look at the final simplified expression and compare it to the given options.

Final Simplified Expression

After combining the like terms in the expression $21 \sqrt[3]{15} - 9 \sqrt[3]{15}$, we arrived at the final simplified expression: $12 \sqrt[3]{15}$. This means that the original expression is equivalent to $12 \sqrt[3]{15}$. To ensure we have the correct answer, let's quickly recap the steps we took: First, we identified that $21 \sqrt[3]{15}$ and $-9 \sqrt[3]{15}$ are like terms because they have the same index (3) and the same radicand (15). Then, we combined these like terms by subtracting their coefficients: $21 - 9 = 12$. Finally, we multiplied the result by the radical part, giving us $12 \sqrt[3]{15}$. This process highlights the importance of recognizing and combining like terms to simplify radical expressions. The simplified form not only makes the expression easier to understand but also reduces the potential for errors in further calculations. In this case, the final simplified expression $12 \sqrt[3]{15}$ is the most concise and clear representation of the original expression. Let's now compare this result with the given options to select the correct answer. This ensures that we have followed the steps accurately and have arrived at the correct solution. Let's move on to identifying the correct option.

Selecting the Correct Option

Now that we have the simplified expression $12 \sqrt[3]{15}$, we need to select the correct option from the choices provided. The question presented the following options:

A. 12 B. $30 \sqrt[3]{15}$ C. $12 \sqrt[3]{5}$ D. $12 \sqrt[3]{15}$

By comparing our simplified expression with the options, we can clearly see that option D, $12 \sqrt[3]{15}$, matches our result. Option A is incorrect because it does not include the radical part. Option B is also incorrect as it has the wrong coefficient. Option C is incorrect because it has the wrong radicand. Therefore, the correct answer is option D. This step is crucial in problem-solving as it confirms that we have accurately followed the necessary steps and arrived at the correct conclusion. Choosing the correct option validates the entire simplification process, from identifying like terms to combining them. In the context of exams and assessments, correctly selecting the answer earns you the marks and demonstrates your understanding of the topic. Let’s summarize the entire process and reinforce the key concepts.

Conclusion

In summary, simplifying the expression $21 \sqrt[3]{15} - 9 \sqrt[3]{15}$ involves a few key steps: identifying like terms, combining the coefficients of those terms, and retaining the radical part. The like terms in this expression are $21 \sqrt[3]{15}$ and $-9 \sqrt[3]{15}$, both having the same index (3) and radicand (15). By subtracting the coefficients (21 - 9), we get 12. Thus, the simplified expression is $12 \sqrt[3]{15}$. This matches option D, which is the correct answer. Understanding how to simplify radical expressions is a fundamental skill in algebra and is essential for tackling more complex mathematical problems. The ability to recognize like terms and correctly combine them allows for efficient and accurate simplification. This process not only helps in solving problems but also enhances overall mathematical comprehension. By mastering these basic techniques, students can build a solid foundation for further studies in mathematics. The key takeaway is that simplifying expressions involves breaking down the problem into manageable steps and applying the appropriate rules and principles. Remember to always look for like terms, combine them accurately, and ensure the final answer is in its simplest form. With practice, simplifying radical expressions becomes a straightforward and manageable task.

The final answer is (D) $12 \sqrt[3]{15}$.