Step-by-Step Guide To Simplifying Complex Mathematical Expressions

Jul 9, 2025 by ADMIN 67 views

Simplifying the Expression: A Step-by-Step Guide

Mathematical expressions can sometimes appear daunting, especially when they involve fractions, parentheses, and multiple operations. However, by following a systematic approach and understanding the order of operations (PEMDAS/BODMAS), even the most complex expressions can be simplified. This article will guide you through the process of simplifying a specific expression, highlighting key concepts and techniques along the way. We'll break down each step, providing clear explanations and ensuring you grasp the underlying principles. Let's dive into simplifying the expression: ${ \frac{2}{7} + \left[ \frac{1}{21} \div \frac{1}{4} - \left( \frac{1}{5} - \frac{1}{8} \right) \times 2 \right] }$

Understanding the Order of Operations (PEMDAS/BODMAS)

Before we begin simplifying the expression, it's crucial to understand the order of operations. This is a set of rules that dictates the sequence in which mathematical operations should be performed. The acronyms PEMDAS and BODMAS are commonly used to remember this order:

Parentheses / Brackets
Exponents / Orders
Multiplication and Division (from left to right)
Addition and Subtraction (from left to right)

By adhering to this order, we ensure that we arrive at the correct simplified answer. Ignoring this order can lead to drastically different and incorrect results. This foundational understanding is essential for tackling any mathematical expression, no matter how complex.

Step 1: Simplifying the Innermost Parentheses

The first step in simplifying our expression is to tackle the innermost parentheses: (1/5 - 1/8). To subtract these fractions, we need a common denominator. The least common multiple of 5 and 8 is 40. Let's rewrite the fractions with this common denominator: ${ \frac{1}{5} = \frac{1 \times 8}{5 \times 8} = \frac{8}{40} }$ ${ \frac{1}{8} = \frac{1 \times 5}{8 \times 5} = \frac{5}{40} }$ Now we can subtract: ${ \frac{8}{40} - \frac{5}{40} = \frac{3}{40} }$ So, the expression within the innermost parentheses simplifies to 3/40. This seemingly small step is crucial, as it sets the stage for further simplification. By systematically addressing the parentheses first, we maintain the correct order of operations and pave the way for accurate calculations in the subsequent steps. This principle of working from the inside out is a cornerstone of simplifying mathematical expressions.

Step 2: Multiplication

Now that we've simplified the innermost parentheses, we can move on to the multiplication operation within the brackets: (3/40) × 2. Multiplying a fraction by a whole number is straightforward. We can rewrite 2 as 2/1 and then multiply the numerators and the denominators: ${ \frac{3}{40} \times 2 = \frac{3}{40} \times \frac{2}{1} = \frac{3 \times 2}{40 \times 1} = \frac{6}{40} }$ We can simplify the fraction 6/40 by dividing both the numerator and the denominator by their greatest common divisor, which is 2: ${ \frac{6}{40} = \frac{6 \div 2}{40 \div 2} = \frac{3}{20} }$ Therefore, the result of the multiplication is 3/20. This step demonstrates the importance of simplifying fractions whenever possible, as it keeps the numbers manageable and reduces the risk of errors in later calculations. By performing the multiplication and simplifying the resulting fraction, we've further streamlined the expression and moved closer to our final answer.

Step 3: Division

Next, we address the division operation within the brackets: (1/21) ÷ (1/4). Dividing by a fraction is the same as multiplying by its reciprocal. The reciprocal of 1/4 is 4/1 (or simply 4). So, we can rewrite the division as multiplication: ${ \frac{1}{21} \div \frac{1}{4} = \frac{1}{21} \times \frac{4}{1} = \frac{1 \times 4}{21 \times 1} = \frac{4}{21} }$ Thus, the division simplifies to 4/21. This step highlights a fundamental concept in fraction arithmetic: dividing by a fraction is equivalent to multiplying by its inverse. By applying this rule, we've transformed a division problem into a multiplication problem, making it easier to solve. Understanding this relationship between division and multiplication is crucial for confidently manipulating fractions in mathematical expressions.

Step 4: Subtraction within the Brackets

Now we can perform the subtraction within the brackets: 4/21 - 3/20. To subtract these fractions, we need a common denominator. The least common multiple of 21 and 20 is 420. Let's rewrite the fractions with this common denominator: ${ \frac{4}{21} = \frac{4 \times 20}{21 \times 20} = \frac{80}{420} }$ ${ \frac{3}{20} = \frac{3 \times 21}{20 \times 21} = \frac{63}{420} }$ Now we can subtract: ${ \frac{80}{420} - \frac{63}{420} = \frac{17}{420} }$ So, the expression within the brackets simplifies to 17/420. Finding a common denominator is a critical skill when adding or subtracting fractions. This step underscores the importance of accurately identifying the least common multiple to ensure correct calculations. By performing the subtraction, we've consolidated the terms within the brackets into a single fraction, further simplifying the overall expression.

Step 5: Addition

Finally, we can perform the addition: 2/7 + 17/420. Again, we need a common denominator. The least common multiple of 7 and 420 is 420. Let's rewrite the fractions with this common denominator: ${ \frac{2}{7} = \frac{2 \times 60}{7 \times 60} = \frac{120}{420} }$ Now we can add: ${ \frac{120}{420} + \frac{17}{420} = \frac{137}{420} }$ Therefore, the simplified expression is 137/420. This final step brings together all the previous simplifications to arrive at the ultimate answer. By adding the fractions with a common denominator, we complete the process of simplifying the expression. The result, 137/420, represents the most concise form of the original, complex expression.

Conclusion

By following the order of operations (PEMDAS/BODMAS) and breaking down the expression into smaller, manageable steps, we successfully simplified the given expression to 137/420. This process demonstrates the power of systematic problem-solving in mathematics. Remember to always prioritize parentheses, exponents, multiplication and division (from left to right), and finally, addition and subtraction (from left to right). With practice, you'll become more confident in simplifying even the most challenging mathematical expressions. The key takeaway is to approach complex problems with a clear strategy and to meticulously execute each step, ensuring accuracy and minimizing errors.

This step-by-step approach not only leads to the correct answer but also fosters a deeper understanding of the underlying mathematical principles involved. Each step builds upon the previous one, creating a logical progression that demystifies the complexity of the original expression. By mastering these techniques, you can confidently tackle a wide range of mathematical problems and develop a strong foundation in arithmetic and algebra.