Simplifying Expressions With Cube Roots An Algebraic Approach
Simplify the expression: $2 a b\left(\sqrt[3]{192 a b^2}\right)-5\left(\sqrt[3]{81 a^4 b^5}\right)$
Introduction
In this article, we will delve into the process of simplifying the algebraic expression 2ab(³√192ab²) - 5(³√81a⁴b⁵). This problem involves working with cube roots and requires a strong understanding of algebraic manipulation and simplification techniques. Our approach will involve breaking down the numbers under the cube roots into their prime factors, identifying perfect cubes, and then extracting these perfect cubes from the radicals. By simplifying each term individually and then combining like terms, we will arrive at the most simplified form of the expression. This step-by-step approach will not only provide the solution but also offer a clear understanding of the methodology involved in simplifying such expressions. Throughout the process, we will emphasize key algebraic rules and properties, ensuring a comprehensive grasp of the concepts involved. Mastering these techniques is essential for anyone studying algebra, as they frequently appear in more complex problems and are fundamental to higher-level mathematics. Let's embark on this mathematical journey and unravel the intricacies of simplifying this expression.
Breaking Down the Expression
To simplify the given expression, 2ab(³√192ab²) - 5(³√81a⁴b⁵), we first need to focus on simplifying the cube roots individually. The key to simplifying cube roots lies in identifying perfect cubes within the radicands (the expressions inside the cube root). To do this effectively, we'll break down the numbers and variables into their prime factors. This allows us to see which factors occur in multiples of three, which can then be extracted from the cube root. For the first term, ³√192ab², we need to factorize 192. The prime factorization of 192 is 2⁶ × 3. This can be rewritten as 2³ × 2³ × 3. Similarly, we analyze the variables a and b². In the second term, ³√81a⁴b⁵, we factorize 81, which is 3⁴, and rewrite it as 3³ × 3. The variables a⁴ can be expressed as a³ × a, and b⁵ can be expressed as b³ × b². By breaking down each component in this manner, we prepare the expression for simplification, making it easier to identify and extract perfect cubes from the cube roots. This meticulous approach is crucial for accurately simplifying radical expressions.
Simplifying the Cube Roots
Now that we have broken down the numbers and variables under the cube roots into their prime factors, we can proceed with simplifying each cube root individually. Starting with the first term, ³√192ab², we recall that 192 can be expressed as 2⁶ × 3, which is the same as 2³ × 2³ × 3. Therefore, ³√192ab² can be written as ³√(2³ × 2³ × 3 × a × b²). We can extract the perfect cubes, which are 2³, from under the cube root. Each pair of 2³ gives us a factor of 2 outside the cube root. This leaves us with 2 × 2 × ³√(3ab²), which simplifies to 4³√(3ab²). Next, we move on to the second term, ³√81a⁴b⁵. We know that 81 is 3⁴, which can be written as 3³ × 3. The term ³√81a⁴b⁵ can thus be rewritten as ³√(3³ × 3 × a³ × a × b³ × b²). Again, we extract the perfect cubes, which are 3³, a³, and b³. This gives us 3ab³√(3ab²). By simplifying each cube root in this manner, we make the overall expression much easier to manage and proceed with further simplification steps. The ability to identify and extract perfect cubes is a fundamental skill in simplifying radical expressions.
Combining Like Terms
After simplifying the cube roots in the expression 2ab(³√192ab²) - 5(³√81a⁴b⁵), we now have 2ab(4³√(3ab²)) - 5(3ab³√(3ab²)). The next step is to multiply the coefficients and the terms outside the cube roots. Multiplying 2ab by 4 in the first term gives us 8ab³√(3ab²). Similarly, multiplying -5 by 3ab in the second term gives us -15ab³√(3ab²). Now, our expression looks like 8ab³√(3ab²) - 15ab³√(3ab²). We can see that both terms have the same cube root part, which is ³√(3ab²), and the same variables outside the cube root, which is ab. This means that they are like terms and can be combined. To combine like terms, we simply add or subtract their coefficients. In this case, we subtract 15 from 8, which gives us -7. Therefore, the simplified expression is -7ab³√(3ab²). This process of combining like terms is a crucial step in simplifying algebraic expressions, as it reduces the expression to its most concise form. The ability to recognize and combine like terms is essential for further algebraic manipulations and problem-solving.
Final Simplified Expression
After carefully simplifying the expression 2ab(³√192ab²) - 5(³√81a⁴b⁵) step by step, we have arrived at the final simplified form. We began by breaking down the numbers and variables under the cube roots into their prime factors, which allowed us to identify and extract perfect cubes. This process transformed the original expression into 2ab(4³√(3ab²)) - 5(3ab³√(3ab²)). We then multiplied the coefficients and terms outside the cube roots, resulting in 8ab³√(3ab²) - 15ab³√(3ab²). Recognizing that these terms were like terms, we combined them by subtracting their coefficients. This led us to the final simplified expression: -7ab³√(3ab²). This final form represents the most concise and simplified version of the original expression. The process we followed demonstrates the importance of understanding prime factorization, perfect cubes, and the rules for simplifying radical expressions. Mastering these techniques is crucial for success in algebra and higher-level mathematics. The journey from the initial complex expression to the final simplified form highlights the power of methodical algebraic manipulation.
Conclusion
In conclusion, we have successfully simplified the algebraic expression 2ab(³√192ab²) - 5(³√81a⁴b⁵) to its final form, -7ab³√(3ab²). This process involved several key steps, including breaking down numbers into their prime factors, identifying and extracting perfect cubes from under the cube root, multiplying coefficients, and combining like terms. Each of these steps is a fundamental technique in algebra and is essential for simplifying more complex expressions. The ability to simplify radical expressions is not only important for academic purposes but also has practical applications in various fields, such as engineering, physics, and computer science. By mastering these techniques, students can develop a deeper understanding of mathematical principles and improve their problem-solving skills. This exercise serves as a valuable example of how a methodical approach and a strong understanding of algebraic rules can lead to the simplification of seemingly complex expressions. The final simplified expression provides a clear and concise representation of the original problem, highlighting the power of algebraic manipulation.