What Is The Simplified Form Of The Expression 7(∛2x) - 3(∛16x) - 3(∛8x)?
This article delves into the process of simplifying radical expressions, using the example:
7(∛2x) - 3(∛16x) - 3(∛8x)
We will explore the steps involved in simplifying this expression, highlighting key concepts and techniques in mathematics. This detailed explanation aims to provide a clear understanding of how to approach similar problems.
Understanding Radical Expressions
Radical expressions involve roots, such as square roots, cube roots, and higher-order roots. Simplifying radical expressions often involves identifying perfect powers within the radicand (the expression under the root) and extracting them. To effectively simplify radical expressions, a solid grasp of root properties and factorization is crucial. The main goal is to reduce the radicand to its simplest form, making the entire expression more manageable and easier to understand. Understanding the properties of radicals, including how to multiply, divide, add, and subtract them, is essential for performing these simplifications accurately. Furthermore, recognizing perfect squares, cubes, and other powers can significantly expedite the simplification process. By mastering these fundamental skills, one can confidently tackle a wide array of problems involving radicals.
For instance, consider the expression √(25x). We know that 25 is a perfect square (5 * 5), so we can simplify this to 5√x. Similarly, for cube roots, we look for perfect cubes. The cube root of 27 (∛27) is 3 because 3 * 3 * 3 = 27. When dealing with more complex expressions, breaking down the radicand into its prime factors can help identify these perfect powers. Additionally, understanding how to combine like terms involving radicals, which means terms with the same radicand and index, is crucial for further simplification. This involves treating the radical part as a common factor and performing the arithmetic operations on the coefficients. Therefore, a comprehensive approach to simplifying radical expressions involves not only knowing the rules but also developing a strategic mindset to identify and apply the most efficient simplification techniques.
Breaking Down the Problem
To simplify the given expression, we need to break down each term and look for perfect cubes within the radicands. The expression is:
7(∛2x) - 3(∛16x) - 3(∛8x)
Let's tackle each term individually.
Term 1: 7(∛2x)
In the first term, 7(∛2x), the radicand is 2x. There are no perfect cubes within 2x, so this term remains as it is. This initial analysis is crucial as it sets the baseline for further simplification. Identifying whether a term can be simplified at the outset prevents unnecessary steps and helps focus on the more complex parts of the expression. The key here is to recognize that both 2 and x are prime factors, and neither can be expressed as a perfect cube. This term serves as a building block, and understanding its irreducible form is essential for combining like terms later in the process. Furthermore, retaining the exact form of this term allows for an accurate comparison with other simplified terms, ensuring that the final answer is both correct and in its most reduced state. Thus, the initial assessment of 7(∛2x) as an irreducible term is a critical step in the overall simplification strategy.
Term 2: -3(∛16x)
In the second term, -3(∛16x), we can break down 16x. 16 can be written as 2^4, which is 2^3 * 2. So, we have:
-3(∛(2^3 * 2 * x))
We can extract the ∛2^3, which is 2. Therefore, the term becomes:
-3 * 2(∛(2x)) = -6(∛2x)
This step demonstrates a crucial aspect of simplifying radicals: identifying and extracting perfect cube factors. By recognizing that 16 is equivalent to 2^4, we can further decompose it into 2^3 * 2, where 2^3 is a perfect cube. This decomposition allows us to take the cube root of 2^3, which is 2, and move it outside the radical. This process significantly reduces the complexity of the radical expression and brings us closer to the simplified form. Moreover, this technique highlights the importance of understanding exponential notation and how it relates to roots. The ability to convert numbers into their prime factorizations and identify perfect powers is a fundamental skill in simplifying radical expressions. The resulting term, -6(∛2x), is now in a form that can be easily combined with other terms that share the same radicand, making further simplification straightforward.
Term 3: -3(∛8x)
In the third term, -3(∛8x), we recognize that 8 is a perfect cube (2^3). So, we have:
-3(∛(2^3 * x))
We can extract the ∛2^3, which is 2. Therefore, the term becomes:
-3 * 2(∛x) = -6(∛x)
This simplification further exemplifies the process of identifying and extracting perfect cube factors from under the radical. Recognizing 8 as 2 cubed (2^3) is the key to simplifying this term. By extracting the cube root of 8, which is 2, we move it outside the radical, leaving us with a simpler radical expression. This step not only reduces the complexity of the term but also highlights the importance of memorizing common perfect cubes and powers. The ability to quickly identify these values can significantly speed up the simplification process. Furthermore, the resulting term, -6(∛x), now has a different radicand than the previous terms, indicating that it cannot be combined with them directly. This distinction underscores the importance of simplifying each term individually before attempting to combine like terms, ensuring that only terms with identical radicands and indices are grouped together. This methodical approach is essential for accurate and efficient simplification of complex radical expressions.
Combining Like Terms
Now that we have simplified each term, we can combine like terms. The simplified expression is:
7(∛2x) - 6(∛2x) - 6(∛x)
We can combine the terms with the same radicand (∛2x):
(7 - 6)(∛2x) - 6(∛x)
This simplifies to:
1(∛2x) - 6(∛x)
Which is:
∛2x - 6(∛x)
The process of combining like terms is a fundamental step in simplifying algebraic expressions, and it is particularly crucial when dealing with radicals. After simplifying individual terms, as demonstrated in the previous sections, the next step involves identifying terms with identical radicands and indices. In this case, 7(∛2x) and -6(∛2x) share the same radicand (2x) and index (cube root), allowing us to combine their coefficients. The coefficients 7 and -6 are combined through subtraction, resulting in 1. This step highlights the principle that only terms with the same radical part can be combined, much like combining like variables in algebraic expressions. The term -6(∛x) cannot be combined with the other terms because it has a different radicand (x). This distinction reinforces the importance of careful observation and accurate identification of like terms. The result, ∛2x - 6(∛x), represents the simplified form of the expression, where all possible combinations have been made, and the expression is now in its most concise and understandable form.
Final Answer
Therefore, the simplified form of the given expression is:
∛2x - 6(∛x)
This corresponds to option C.
Conclusion
Simplifying radical expressions involves breaking down the radicands, identifying perfect powers, extracting roots, and combining like terms. By following these steps systematically, we can efficiently simplify complex expressions and arrive at the correct answer. This detailed walkthrough has provided a comprehensive understanding of the process, enabling readers to apply these techniques to similar problems with confidence. Mastering the simplification of radical expressions is a valuable skill in mathematics, paving the way for more advanced topics and problem-solving scenarios.