Simplifying Radical Expressions A Step By Step Guide

Introduction: Delving into Radical Expressions

At the heart of algebraic manipulations lies the simplification of expressions, especially those involving radicals. Understanding radicals and their properties is crucial for success in various mathematical domains. In this comprehensive exploration, we aim to dissect and simplify the expression $3 b^2(\sqrt[3]{54 a})+3(\sqrt[3]{2 a b^6})$, offering a step-by-step guide to arrive at the most simplified form. This process requires not only a firm grasp of algebraic principles but also an understanding of how to manipulate radicals effectively. We will cover key concepts such as factoring, extracting perfect cubes, and combining like terms, ensuring a clear and thorough understanding of each step. Our goal is to transform what might seem like a complex expression into a straightforward, manageable form, demonstrating the elegance and power of algebraic simplification.

Deconstructing the Expression: A Step-by-Step Approach

To effectively tackle the expression $3 b^2(\sqrt[3]{54 a})+3(\sqrt[3]{2 a b^6})$, we must break it down into manageable components. The key here is to identify and simplify the radicals by factoring out perfect cubes. Let's begin by examining each term separately, focusing on simplifying the cube roots. This methodical approach will allow us to unveil the underlying structure of the expression, making it easier to identify opportunities for simplification. The first term, $3 b^2(\sqrt[3]{54 a})$, contains the cube root of $54a$. We'll need to factor 54 to find any perfect cube factors. Similarly, the second term, $3(\sqrt[3]{2 a b^6})$, involves the cube root of $2ab^6$, where we'll need to consider both the numerical and variable parts. By addressing each term individually, we can maintain clarity and reduce the likelihood of errors, ensuring a successful simplification.

Simplifying the First Term: $3 b^2(\sqrt[3]{54 a})$

Our initial focus is on simplifying the first term, $3 b^2(\sqrt[3]{54 a})$. The central task here is to simplify the cube root of $54a$. To do this, we need to identify the largest perfect cube that divides 54. The prime factorization of 54 is $2 \times 3^3$. This reveals that 27, or $3^3$, is a perfect cube factor of 54. Thus, we can rewrite $\sqrt[3]{54 a}$ as $\sqrt[3]{27 \times 2 a}$. Now, we can use the property of radicals that allows us to separate the cube root of a product into the product of cube roots: $\sqrt[3]{27 \times 2 a} = \sqrt[3]{27} \times \sqrt[3]{2 a}$. Since the cube root of 27 is 3, we simplify further to $3 \sqrt[3]{2 a}$. Substituting this back into the first term, we get $3 b^2 (3 \sqrt[3]{2 a})$. Finally, multiplying the constants gives us $9 b^2 \sqrt[3]{2 a}$. This step-by-step breakdown illustrates how identifying and extracting perfect cubes is crucial for simplifying radical expressions. The ability to recognize these patterns allows us to transform complex radicals into more manageable forms, paving the way for further simplification.

Simplifying the Second Term: $3(\sqrt[3]{2 a b^6})$

Now, let's turn our attention to the second term: $3(\sqrt[3]{2 a b^6})$. Here, we aim to simplify the cube root of $2ab^6$. This term presents an interesting opportunity as it includes a variable raised to a power that is a multiple of 3, specifically $b^6$. We can rewrite $b^6$ as $(b^2)^3$, which makes it clear that $b^6$ is a perfect cube. Therefore, we can express the cube root of $2ab^6$ as $\sqrt[3]{2 a (b^2)^3}$. Using the property of radicals, we separate the cube root of the product into the product of cube roots: $\sqrt[3]{2 a (b^2)^3} = \sqrt[3]{(b^2)^3} \times \sqrt[3]{2 a}$. The cube root of $(b^2)^3$ is simply $b^2$, so we have $b^2 \sqrt[3]{2 a}$. Substituting this back into the second term gives us $3(b^2 \sqrt[3]{2 a})$, which simplifies to $3 b^2 \sqrt[3]{2 a}$. This simplification highlights the importance of recognizing perfect cube powers within variables, allowing us to extract them from the radical and further simplify the expression. By isolating and addressing these perfect cubes, we can transform complex terms into more manageable components.

Combining Like Terms: Bringing it All Together

With both terms simplified, we now have $9 b^2 \sqrt[3]{2 a}$ and $3 b^2 \sqrt[3]{2 a}$. The next step is to combine these like terms. Notice that both terms share the same radical part, which is $b^2 \sqrt[3]{2 a}$. This means we can treat $b^2 \sqrt[3]{2 a}$ as a common factor and combine the coefficients. To do this, we simply add the coefficients of the two terms: $9 + 3 = 12$. Thus, the combined term is $12 b^2 \sqrt[3]{2 a}$. This process of combining like terms is a fundamental algebraic technique that allows us to consolidate expressions and arrive at the most simplified form. By recognizing and grouping terms with identical variable and radical parts, we can streamline the expression and reveal its underlying simplicity. The ability to identify and combine like terms is a crucial skill in algebra, enabling us to manipulate and simplify expressions effectively.

The Final Simplified Expression: $12 b^2(\sqrt[3]{2 a})$

After simplifying each term individually and then combining like terms, we arrive at the final simplified expression: $12 b^2(\sqrt[3]{2 a})$. This result demonstrates the power of algebraic manipulation and simplification techniques. By systematically breaking down the original expression, identifying perfect cubes, and combining like terms, we transformed a seemingly complex expression into a much simpler and more manageable form. This process not only provides a solution to the problem but also reinforces the importance of understanding the underlying principles of algebra. The ability to simplify radical expressions is a valuable skill in mathematics, with applications in various fields, including calculus, physics, and engineering. The simplified expression, $12 b^2(\sqrt[3]{2 a})$, is the most concise and elegant representation of the original sum, showcasing the beauty and efficiency of mathematical simplification.

Conclusion: Mastering Algebraic Simplification

In conclusion, the simplification of the expression $3 b^2(\sqrt[3]{54 a})+3(\sqrt[3]{2 a b^6})$ to $12 b^2(\sqrt[3]{2 a})$ showcases the power and importance of algebraic manipulation. This journey involved several key steps, including factoring, extracting perfect cubes from radicals, and combining like terms. Each step required a solid understanding of algebraic principles and the properties of radicals. The process not only simplified the expression but also provided a deeper insight into the underlying mathematical structure. Mastering algebraic simplification is a fundamental skill in mathematics, opening doors to more advanced concepts and applications. The ability to transform complex expressions into simpler forms is essential for problem-solving in various fields. By understanding and applying these techniques, students and practitioners alike can confidently tackle algebraic challenges and unlock the beauty and elegance of mathematics. The final simplified expression, $12 b^2(\sqrt[3]{2 a})$, stands as a testament to the effectiveness of these methods and the importance of algebraic proficiency.

Therefore, the correct answer is B. $12 b^2(\sqrt[3]{2 a})$.