Solving (b) √[3]{6859} / √[4]{1296} × 3/57 × 42 A Step-by-Step Guide

JU07/15/2025 08, 2025 by THE IDEN 69 views

Unraveling the Mathematical Equation: A Step-by-Step Guide to Solving (b) √[3]{6859} / √[4]{1296} × 3/57 × 42

Navigating the world of mathematics often involves tackling intricate equations that require a methodical approach. This article delves into the step-by-step solution of a seemingly complex expression, providing clarity and understanding along the way. Our focus is on the equation: (b) √[3]{6859} / √[4]{1296} × 3/57 × 42. By dissecting each component and applying the correct mathematical operations, we will arrive at the final answer, illustrating the beauty and precision of mathematical problem-solving. This article aims to enhance your understanding of mathematical operations and provide a clear, concise method for solving similar problems. We begin by isolating each part of the equation, simplifying it, and then combining the results. This methodical approach ensures accuracy and clarity, making the complex seem simple.

1. Understanding the Components of the Equation

To effectively solve the equation (b) √[3]{6859} / √[4]{1296} × 3/57 × 42, it's crucial to first understand each component individually. This involves recognizing the different mathematical operations and the values they operate on. Let's break down the equation into its core elements:

Cube Root of 6859 (√[3]{6859}): This component requires us to find a number that, when multiplied by itself three times, equals 6859.
Fourth Root of 1296 (√[4]{1296}): Here, we need to identify a number that, when multiplied by itself four times, results in 1296.
Fraction 3/57: This is a simple fraction representing a part of a whole. It can be simplified further to its lowest terms.
Integer 42: This is a whole number that will be multiplied with the result of the other operations.

By understanding these individual components, we can approach the equation in a structured manner, simplifying each part before combining them. This methodical approach is key to solving complex mathematical problems accurately and efficiently. The importance of understanding each component cannot be overstated; it forms the foundation upon which the entire solution is built. Ignoring this preliminary step can lead to confusion and errors later on. Therefore, take the time to dissect and comprehend each element before moving forward. Each component is like a piece of a puzzle, and only when they are understood can they be properly assembled to reveal the complete picture. Moreover, this understanding allows for the application of relevant mathematical principles and rules, ensuring that the solution is not only correct but also logically sound. It allows us to apply the correct order of operations and ensures that we don't make common mistakes, such as simplifying incorrectly. This foundational understanding sets the stage for a smooth and accurate problem-solving process, ultimately leading to the correct answer and a deeper appreciation of mathematical concepts. By breaking down the equation into manageable parts, we not only simplify the task at hand but also gain a clearer perspective on the overall structure of the problem.

2. Simplifying the Cube Root of 6859

Finding the cube root of 6859 (√[3]{6859}) is a critical step in solving the overall equation. The cube root of a number is a value that, when multiplied by itself three times, equals the original number. To find √[3]{6859}, we need to determine this value. One way to approach this is through prime factorization or by recognizing perfect cubes. In this case, 6859 is a perfect cube. The number 19, when multiplied by itself three times (19 × 19 × 19), equals 6859. Therefore, the cube root of 6859 is 19. This simplification significantly reduces the complexity of the equation, making it easier to work with. The ability to recognize and simplify roots, whether cube roots, square roots, or higher-order roots, is a fundamental skill in mathematics. It not only simplifies equations but also provides a deeper understanding of numerical relationships. This skill is particularly useful in algebra and calculus, where complex expressions often need to be simplified before further operations can be performed. Understanding perfect cubes and their roots is an invaluable tool in any mathematical endeavor. Moreover, the process of finding a cube root can be visualized as finding the side length of a cube given its volume, which further cements the concept in a tangible way. In practical terms, simplifying cube roots allows for more efficient calculations and can prevent errors that might arise from working with larger, unsimplified numbers. The simplification also allows us to see the equation in a clearer light, making subsequent steps more straightforward. By reducing the cube root of 6859 to 19, we've transformed a complex operation into a simple numerical value, paving the way for the next steps in solving the equation.

3. Calculating the Fourth Root of 1296

Next in our equation, we encounter the fourth root of 1296 (√[4]{1296}). Similar to the cube root, the fourth root of a number is the value that, when multiplied by itself four times, equals the original number. To find √[4]{1296}, we must identify this value. We can approach this by prime factorization or by recognizing perfect fourth powers. The number 1296 is a perfect fourth power. The number 6, when multiplied by itself four times (6 × 6 × 6 × 6), equals 1296. Thus, the fourth root of 1296 is 6. This simplification is another crucial step in making the equation more manageable. Understanding and simplifying roots, such as fourth roots, is an essential skill in algebra and number theory. It allows for the efficient manipulation of expressions and equations, making complex problems more tractable. This skill is particularly valuable in fields such as engineering and physics, where equations often involve higher-order roots and exponents. The process of finding the fourth root can be thought of as finding the side length of a tesseract (a four-dimensional hypercube) given its hypervolume. This analogy helps to visualize the concept and make it more intuitive. In practical applications, simplifying fourth roots, or any higher-order roots, can greatly reduce computational complexity. It allows for quicker and more accurate calculations, which is crucial in time-sensitive situations. Furthermore, by reducing the fourth root of 1296 to 6, we maintain the integrity of the equation while making it significantly easier to solve. This simplification not only simplifies the calculation but also enhances our understanding of the numerical relationships within the equation. Simplifying complex components like this one helps to maintain clarity and accuracy, which are paramount in mathematical problem-solving.

4. Simplifying the Fraction 3/57

Another component of our equation is the fraction 3/57. Simplifying fractions is a fundamental arithmetic skill that involves reducing the fraction to its lowest terms. This means finding the greatest common divisor (GCD) of the numerator and the denominator and dividing both by that GCD. In the case of 3/57, the GCD of 3 and 57 is 3. Dividing both the numerator and the denominator by 3, we get: (3 ÷ 3) / (57 ÷ 3) = 1/19. Therefore, the simplified form of the fraction 3/57 is 1/19. This simplification makes the subsequent multiplication easier and more straightforward. Simplifying fractions not only makes calculations easier but also provides a clearer understanding of the proportional relationship between the numerator and the denominator. It allows us to see the fraction in its most basic form, which can be helpful in various mathematical contexts. This skill is particularly useful in algebra, calculus, and other advanced mathematical fields, where simplification is often necessary to solve complex problems. Simplifying fractions is also a practical skill that is used in everyday life, such as in cooking, measuring, and financial calculations. The ability to quickly simplify fractions can save time and prevent errors in various situations. In the context of our equation, simplifying 3/57 to 1/19 streamlines the calculations and ensures accuracy. It also highlights the importance of recognizing and applying basic arithmetic principles in solving more complex problems. Simplification is a key strategy in mathematics, and mastering it is crucial for success in various mathematical endeavors. By reducing the fraction to its simplest form, we reduce the chances of error and make the overall solution more elegant and efficient.

5. Combining the Simplified Components

Now that we have simplified each component of the equation (b) √[3]{6859} / √[4]{1296} × 3/57 × 42, we can combine them to find the final result. We have:

√[3]{6859} = 19
√[4]{1296} = 6
3/57 = 1/19

Substituting these simplified values back into the equation, we get: 19 / 6 × 1/19 × 42. To solve this, we perform the operations in the order they appear from left to right, following the standard order of operations (PEMDAS/BODMAS). First, we divide 19 by 6: 19 / 6 ≈ 3.1667. Next, we multiply this result by 1/19: 3. 1667 × 1/19 ≈ 0.1667. Finally, we multiply this result by 42: 0.1667 × 42 ≈ 7. Therefore, the final answer to the equation is approximately 7. Combining simplified components is a crucial step in solving complex equations. It allows us to work with smaller, more manageable numbers, reducing the likelihood of errors. This process also highlights the interconnectedness of different mathematical operations and the importance of understanding how they interact with each other. In this case, simplifying the roots and the fraction before performing the multiplication and division made the calculation much easier. The order in which we perform these operations is also critical, as changing the order can lead to a different result. Following the order of operations ensures that we arrive at the correct answer. Furthermore, this process demonstrates the power of simplification in mathematics. By breaking down a complex problem into smaller, simpler parts, we can solve it more efficiently and accurately. This approach is applicable not only to mathematical problems but also to various other fields, such as engineering, physics, and computer science. By methodically combining the simplified components, we have successfully solved the equation and demonstrated the effectiveness of a structured approach to problem-solving. This final step brings together all the previous simplifications, resulting in a clear and concise answer, which showcases the beauty and precision of mathematical calculations.

6. The Final Solution

After simplifying each component of the equation and combining the results, we arrive at the final solution. The equation (b) √[3]{6859} / √[4]{1296} × 3/57 × 42 simplifies to approximately 7. This result is obtained by first finding the cube root of 6859, which is 19, and the fourth root of 1296, which is 6. The fraction 3/57 is simplified to 1/19. Substituting these values into the equation gives us 19 / 6 × 1/19 × 42. Performing the operations in the correct order leads to the final answer of approximately 7. This solution underscores the importance of methodical problem-solving in mathematics. By breaking down the equation into smaller, more manageable parts, we can simplify the process and reduce the risk of errors. Each step, from simplifying the roots to reducing the fraction, plays a crucial role in reaching the correct answer. The final solution not only provides a numerical result but also demonstrates the power of mathematical principles and techniques. It highlights the importance of understanding roots, fractions, and the order of operations. Moreover, this exercise serves as a valuable learning experience, reinforcing the skills necessary for tackling more complex mathematical problems in the future. The clarity and precision of the final answer are a testament to the rigor and elegance of mathematical reasoning. In conclusion, the journey from the complex equation to the final solution illustrates the effectiveness of a structured and methodical approach. The final answer, approximately 7, stands as a testament to the power of mathematical simplification and problem-solving.

In conclusion, solving complex mathematical expressions like (b) √[3]{6859} / √[4]{1296} × 3/57 × 42 requires a methodical and step-by-step approach. By breaking down the equation into its individual components, simplifying each part, and then combining the results, we can arrive at the final answer with confidence. This process not only provides the solution but also enhances our understanding of mathematical principles and techniques. The key takeaways from this exercise include the importance of understanding roots, simplifying fractions, and following the correct order of operations. These skills are fundamental to success in mathematics and can be applied to a wide range of problems. Moreover, the ability to approach complex problems in a structured manner is a valuable skill that extends beyond mathematics and into various other fields. By mastering these techniques, we can unlock the power of mathematical problem-solving and gain a deeper appreciation for the beauty and precision of mathematics. This comprehensive approach not only solves the problem at hand but also equips us with the tools and knowledge to tackle future challenges. The process of simplifying and combining components is a powerful strategy in mathematics, and it is a skill that can be honed with practice and dedication. The final solution, approximately 7, is not just a number; it is the culmination of a series of logical steps and mathematical operations, demonstrating the effectiveness of our approach. This exercise serves as a testament to the power of methodical problem-solving and the importance of mastering fundamental mathematical skills. In summary, the journey from the initial equation to the final solution underscores the value of a structured approach and the significance of mathematical understanding.