Simplifying The Algebraic Expression 2x³y - [ 5/6x³y - 2/3x³y - ( 4/7x³y - 1/2x³y - 3/14x³y ) + 5/21x³y ] - 25/21x³y

Jul 9, 2025 by ADMIN 117 views

Simplify Algebraic Expression 2x³y - [ 5/6x³y - 2/3x³y - ( 4/7x³y - 1/2x³y - 3/14x³y ) + 5/21x³y ] - 25/21x³y

Introduction

In the realm of mathematics, algebraic expressions form the bedrock of numerous equations and formulas. Simplifying these expressions is a fundamental skill, crucial for solving complex problems and gaining a deeper understanding of mathematical concepts. This article delves into the process of simplifying a given algebraic expression, providing a step-by-step guide to unraveling its intricacies. Our focus will be on the expression: 2x³y - [ 5/6x³y - 2/3x³y - ( 4/7x³y - 1/2x³y - 3/14x³y ) + 5/21x³y ] - 25/21x³y. This expression, while seemingly complex, can be simplified through careful application of algebraic principles and techniques.

This comprehensive exploration will not only demonstrate the simplification process but also emphasize the underlying mathematical principles at play. By breaking down each step and providing clear explanations, we aim to empower readers with the ability to tackle similar algebraic challenges with confidence. Whether you are a student seeking to enhance your understanding or a seasoned mathematician looking for a refresher, this article offers valuable insights into the art of algebraic simplification.

Algebraic simplification is more than just a mathematical exercise; it's a powerful tool that enhances problem-solving skills and logical reasoning. By mastering the techniques presented here, you'll be well-equipped to navigate a wide range of mathematical problems, from basic equations to advanced calculus. So, let's embark on this journey of simplification, where we'll transform a seemingly complex expression into its most elegant and concise form.

Understanding the Expression

Before we dive into the simplification process, it's crucial to understand the structure of the given algebraic expression. The expression is: 2x³y - [ 5/6x³y - 2/3x³y - ( 4/7x³y - 1/2x³y - 3/14x³y ) + 5/21x³y ] - 25/21x³y. It involves terms with the variables x and y, both raised to specific powers. Specifically, we have the term x³y appearing throughout the expression. The coefficients of these terms are fractions, which adds a layer of complexity. The expression also includes nested parentheses and brackets, indicating the order of operations we need to follow. The presence of subtraction and addition operations further dictates how we combine the terms.

To effectively simplify this expression, we'll need to adhere to the order of operations, commonly remembered by the acronym PEMDAS/BODMAS (Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction). This means we'll first address the innermost parentheses, then move outwards, simplifying the brackets, and finally combining the terms outside the brackets. Understanding this hierarchical structure is paramount for accurate simplification.

The expression's complexity stems from the fractional coefficients and the nested grouping symbols. However, by systematically applying the order of operations and combining like terms, we can reduce it to a more manageable form. Recognizing the common term x³y throughout the expression is also key, as it allows us to focus on the coefficients and perform arithmetic operations on them. This initial assessment sets the stage for a methodical simplification process, ensuring we address each component in the correct sequence.

Step-by-Step Simplification

To simplify the algebraic expression 2x³y - [ 5/6x³y - 2/3x³y - ( 4/7x³y - 1/2x³y - 3/14x³y ) + 5/21x³y ] - 25/21x³y, we will follow a step-by-step approach, meticulously addressing each component of the expression. This method ensures accuracy and clarity throughout the simplification process.

Step 1: Simplify the Innermost Parentheses

We begin by tackling the innermost parentheses: ( 4/7x³y - 1/2x³y - 3/14x³y ). To combine these terms, we need to find a common denominator for the fractions 4/7, 1/2, and 3/14. The least common denominator (LCD) for 7, 2, and 14 is 14. We rewrite each fraction with the denominator 14:

4/7 = 8/14
1/2 = 7/14
3/14 remains as 3/14

Now we can combine the terms:

( 8/14x³y - 7/14x³y - 3/14x³y ) = ( 8/14 - 7/14 - 3/14 )x³y = ( -2/14 )x³y = ( -1/7 )x³y

Thus, the innermost parentheses simplify to -1/7x³y.

Step 2: Simplify the Brackets

Next, we address the brackets: [ 5/6x³y - 2/3x³y - ( -1/7x³y ) + 5/21x³y ]. We've already simplified the innermost parentheses, so we substitute the result into the brackets:

[ 5/6x³y - 2/3x³y + 1/7x³y + 5/21x³y ]

Notice that subtracting a negative term becomes addition. Now, we need to find a common denominator for the fractions 5/6, 2/3, 1/7, and 5/21. The LCD for 6, 3, 7, and 21 is 42. We rewrite each fraction with the denominator 42:

5/6 = 35/42
2/3 = 28/42
1/7 = 6/42
5/21 = 10/42

Now we can combine the terms:

( 35/42x³y - 28/42x³y + 6/42x³y + 10/42x³y ) = ( 35/42 - 28/42 + 6/42 + 10/42 )x³y = ( 23/42 )x³y

Thus, the brackets simplify to 23/42x³y.

Step 3: Simplify the Entire Expression

Finally, we simplify the entire expression:

2x³y - [ 23/42x³y ] - 25/21x³y

Substitute the simplified brackets:

2x³y - 23/42x³y - 25/21x³y

To combine these terms, we need to find a common denominator for 1 (the coefficient of 2x³y), 42, and 21. The LCD for 1, 42, and 21 is 42. We rewrite each fraction with the denominator 42:

2 = 84/42
23/42 remains as 23/42
25/21 = 50/42

Now we can combine the terms:

( 84/42x³y - 23/42x³y - 50/42x³y ) = ( 84/42 - 23/42 - 50/42 )x³y = ( 11/42 )x³y

Therefore, the simplified expression is 11/42x³y.

Detailed Breakdown of Each Step

In the preceding section, we presented a step-by-step simplification of the algebraic expression. Here, we will delve deeper into each step, providing a detailed explanation of the mathematical principles and techniques employed. This granular analysis will enhance your understanding and equip you with the skills to tackle similar problems.

Step 1: Simplifying the Innermost Parentheses - A Closer Look

The first step in our simplification journey was to address the innermost parentheses: ( 4/7x³y - 1/2x³y - 3/14x³y ). This step highlights the importance of the order of operations, which dictates that expressions within parentheses are simplified first. To combine these terms, we recognized that they are like terms, all containing the variable part x³y. This allows us to focus on the coefficients – the fractions 4/7, 1/2, and 3/14.

The key to combining fractions is to find a common denominator. The least common denominator (LCD) is the smallest multiple that all the denominators share. In this case, the denominators are 7, 2, and 14. The LCD for these numbers is 14. We then converted each fraction to an equivalent fraction with the denominator 14. This involves multiplying both the numerator and the denominator of each fraction by a suitable factor. For example, to convert 4/7 to an equivalent fraction with a denominator of 14, we multiplied both the numerator and the denominator by 2, resulting in 8/14.

Once we had all the fractions with a common denominator, we could combine them by adding or subtracting the numerators while keeping the denominator the same. This process yielded the simplified fraction -2/14, which we further reduced to -1/7. Finally, we multiplied this simplified fraction by the variable part x³y to obtain the simplified term -1/7x³y. This detailed breakdown illustrates the fundamental principles of fraction manipulation and the importance of finding a common denominator when combining fractions.

Step 2: Simplifying the Brackets - A Comprehensive Explanation

Having simplified the innermost parentheses, we moved on to the brackets: [ 5/6x³y - 2/3x³y - ( -1/7x³y ) + 5/21x³y ]. This step showcases the significance of careful substitution and the rules of arithmetic operations. We began by substituting the simplified expression from the previous step, -1/7x³y, into the brackets. A crucial observation here is the subtraction of a negative term, which transforms into addition. This is a common point of error, and understanding this rule is vital for accurate simplification.

Next, we encountered another set of fractions with different denominators: 5/6, 2/3, 1/7, and 5/21. As before, we needed to find the least common denominator (LCD) to combine these fractions. The LCD for 6, 3, 7, and 21 is 42. We then converted each fraction to an equivalent fraction with the denominator 42, using the same method as in the previous step. For instance, to convert 5/6 to an equivalent fraction with a denominator of 42, we multiplied both the numerator and the denominator by 7, resulting in 35/42.

With all the fractions sharing a common denominator, we combined them by adding and subtracting the numerators while preserving the denominator. This arithmetic operation led to the simplified fraction 23/42. Finally, we multiplied this simplified fraction by the variable part x³y to obtain the simplified term 23/42x³y. This detailed explanation underscores the importance of accurate substitution, the rules of arithmetic operations, and the consistent application of the LCD method when combining fractions.

Step 3: Simplifying the Entire Expression - The Final Touches

The final step in our simplification process involved addressing the entire expression: 2x³y - [ 23/42x³y ] - 25/21x³y. This step emphasizes the culmination of all previous simplifications and the final combination of like terms. We began by substituting the simplified expression from the brackets, 23/42x³y, into the main expression. This substitution effectively removed the brackets, leaving us with a simpler expression to work with.

We then encountered a final set of fractions, including the whole number 2, which we treated as a fraction with a denominator of 1. The fractions were 2/1, 23/42, and 25/21. To combine these fractions, we once again needed to find the least common denominator (LCD). The LCD for 1, 42, and 21 is 42. We converted each fraction to an equivalent fraction with the denominator 42. For example, to convert 2/1 to an equivalent fraction with a denominator of 42, we multiplied both the numerator and the denominator by 42, resulting in 84/42.

With all the fractions sharing a common denominator, we performed the final addition and subtraction operations on the numerators, keeping the denominator constant. This arithmetic process yielded the simplified fraction 11/42. Lastly, we multiplied this simplified fraction by the variable part x³y to arrive at the final simplified term 11/42x³y. This comprehensive explanation highlights the importance of accurate substitution, consistent application of the LCD method, and the careful execution of arithmetic operations to achieve the final simplified expression.

Conclusion

In this comprehensive exploration, we have successfully simplified the algebraic expression 2x³y - [ 5/6x³y - 2/3x³y - ( 4/7x³y - 1/2x³y - 3/14x³y ) + 5/21x³y ] - 25/21x³y to its simplest form: 11/42x³y. This journey through simplification has not only provided a solution but also illuminated the underlying mathematical principles and techniques involved. We meticulously dissected each step, from simplifying the innermost parentheses to combining like terms in the final expression. This methodical approach underscores the importance of precision and attention to detail in algebraic manipulations.

The process began with a clear understanding of the order of operations, encapsulated by the acronym PEMDAS/BODMAS, which guided us through the nested structure of the expression. We then tackled the innermost parentheses, employing the fundamental principles of fraction manipulation, including finding the least common denominator (LCD) and converting fractions to equivalent forms. As we moved outwards, we encountered the need for accurate substitution and the careful application of arithmetic operations, particularly the rule that subtracting a negative term becomes addition.

Each step was meticulously explained, highlighting the rationale behind every action and the mathematical principles at play. This detailed breakdown aimed to foster a deeper understanding of the simplification process, empowering readers to confidently tackle similar algebraic challenges. By mastering these techniques, one can not only simplify complex expressions but also enhance their problem-solving skills and logical reasoning abilities. The final simplified expression, 11/42x³y, stands as a testament to the power of systematic simplification and the beauty of mathematical elegance.

Algebraic simplification is a cornerstone of mathematical proficiency, essential for solving equations, simplifying formulas, and gaining a deeper appreciation for the interconnectedness of mathematical concepts. The journey we've undertaken in this article serves as a valuable guide, providing the tools and insights necessary to navigate the world of algebraic expressions with confidence and skill.