Simplifying Complex Mathematical Expressions A Step-by-Step Guide
This article dives deep into the step-by-step simplification of a complex mathematical expression. We'll break down each component, explaining the order of operations and the reasoning behind each step. This detailed walkthrough aims to enhance understanding and provide a clear path to the solution. Complex mathematical expressions often appear daunting, but by systematically applying the rules of arithmetic and algebra, we can simplify them into manageable parts. The key is to follow the order of operations (PEMDAS/BODMAS) meticulously and pay close attention to the signs and exponents involved. This article serves as a comprehensive guide to navigating such expressions, offering insights into the strategies and techniques required for successful simplification. Let's embark on this mathematical journey together, unraveling the intricacies and arriving at the final solution with clarity and confidence.
Breaking Down the Numerator
The numerator of the expression presents a series of arithmetic operations that must be performed in the correct order. Our first step involves simplifying the expression within the brackets. Inside the brackets, we encounter a combination of addition, multiplication, and subtraction. According to the order of operations, we first address the multiplication: 2 * 4 = 8. Next, we perform the addition: 2 + 8 = 10. Now, we can substitute this value back into the original expression: 5(10) + (6 - 3 + 3). The next step involves simplifying the remaining expression within the parentheses. We perform the multiplication: 5 * 10 = 50. Then, we simplify the second set of parentheses: 6 - 3 + 3 = 6. Now, we substitute these values back into the expression: 50 + 6. Adding these values together, we get 56. Next, we multiply 56 by 3 and then by 2, which yields 336. We then subtract this from 100 and add 5, resulting in -231. This meticulous step-by-step approach ensures that we accurately simplify the numerator, avoiding common errors that can arise from overlooking the order of operations. Each operation is carefully considered and executed, leading us closer to the final solution. This process exemplifies the importance of precision and attention to detail in mathematical problem-solving.
Simplifying the Denominator
The denominator of the expression also requires careful simplification, following the order of operations. Our primary focus here is to address the exponents and the operations within the parentheses. We begin by evaluating the expression (3 + 4 - 2)^2. First, we perform the operations inside the parentheses: 3 + 4 - 2 = 5. Then, we square the result: 5^2 = 25. Next, we focus on the second term in the denominator: 4(12 * 2^12 - 4 * 5) * 1^8. We start by evaluating the exponent: 2^12 = 4096. Then, we perform the multiplication: 12 * 4096 = 49152. Next, we evaluate 4 * 5 = 20. Substituting these values back into the expression, we have: 4(49152 - 20) * 1^8. Now, we subtract 20 from 49152: 49152 - 20 = 49132. We also evaluate 1^8 = 1. Next, we multiply 49132 by 4: 4 * 49132 = 196528. Finally, we multiply by 1, which remains 196528. Now, we can substitute these values back into the denominator: 25 - 196528. Subtracting these values, we get -196503. The simplification of the denominator highlights the importance of handling exponents and large numbers with care. Each step is executed methodically, ensuring that we arrive at the correct value. This thorough approach mirrors the precision required in various mathematical and scientific calculations.
Putting It All Together
Now that we have simplified both the numerator and the denominator, we can combine them to obtain the final result. Our calculated numerator is -231, and the simplified denominator is -196503. Therefore, the fraction becomes -231 / -196503. We can further simplify this fraction by finding the greatest common divisor (GCD) of the numerator and the denominator. The GCD of 231 and 196503 is 3. Dividing both the numerator and the denominator by 3, we get: -231 / 3 = -77 and -196503 / 3 = -65501. Thus, the simplified fraction is -77 / -65501. Since both the numerator and the denominator are negative, we can further simplify by canceling the negative signs, resulting in 77 / 65501. This final fraction represents the simplified form of the original complex expression. The process of combining the simplified numerator and denominator demonstrates how breaking down a complex problem into smaller, manageable parts can lead to a clear and concise solution. The use of the GCD further refines the result, presenting it in its simplest form. This approach underscores the power of systematic simplification in mathematical problem-solving.
Addressing the Second Expression
The second expression provided, 1 / (0^2 - 35 * 10 + 07 * 2 + 5), introduces another set of arithmetic operations that require careful handling. Our first step involves evaluating the terms in the denominator. We start with the exponent: 0^2 = 0. Then, we perform the multiplication: 35 * 10 = 350. Next, we evaluate 07 * 2, which is equivalent to 7 * 2 = 14. Now, we can substitute these values back into the expression: 0 - 350 + 14 + 5. Performing the subtraction and addition, we get: -350 + 14 + 5 = -331. Therefore, the denominator simplifies to -331. Now, the entire expression becomes 1 / -331. This is the simplified form of the second expression. The key to simplifying this expression lies in accurately performing the arithmetic operations in the denominator. Each step, from evaluating the exponent to performing the multiplication and addition, contributes to the final result. This careful approach ensures that we avoid errors and arrive at the correct simplified form.
Analyzing the Third Expression
The third expression, (5^2 - 4 + 4 * 1^8), presents another opportunity to apply the order of operations. Our initial focus is on evaluating the exponent: 5^2 = 25. Next, we evaluate 1^8 = 1. Then, we perform the multiplication: 4 * 1 = 4. Now, we can substitute these values back into the expression: 25 - 4 + 4. Performing the subtraction and addition, we get: 25 - 4 + 4 = 25. Therefore, the simplified form of the third expression is 25. This simplification process underscores the importance of adhering to the order of operations. By systematically addressing each operation, we arrive at the correct result. The expression serves as a concise example of how a complex-looking combination of operations can be simplified through careful application of mathematical rules.
Evaluating the Simplified Expression
The simplified expression, (10^2 - 3 * 50 * 2 + 5) / -300, now needs to be evaluated to obtain its numerical value. Our first step is to evaluate the exponent: 10^2 = 100. Then, we perform the multiplication: 3 * 50 * 2 = 300. Now, we can substitute these values back into the expression: (100 - 300 + 5) / -300. Performing the subtraction and addition in the numerator, we get: 100 - 300 + 5 = -195. So, the expression becomes -195 / -300. We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor (GCD), which is 15. Dividing both by 15, we get: -195 / 15 = -13 and -300 / 15 = -20. Thus, the simplified fraction is -13 / -20. Since both the numerator and the denominator are negative, we can cancel the negative signs, resulting in 13 / 20. This final fraction, 13/20, represents the evaluated form of the given expression. The process of evaluating this expression demonstrates how simplifying and then combining terms can lead to a manageable calculation. The use of the GCD further refines the result, presenting it in its simplest form.
In conclusion, simplifying complex mathematical expressions requires a systematic approach, meticulous attention to detail, and a firm understanding of the order of operations. By breaking down expressions into smaller parts, addressing exponents, and performing arithmetic operations in the correct sequence, we can navigate even the most daunting calculations with confidence. The examples presented in this article illustrate the power of this approach, providing a clear roadmap for tackling similar mathematical challenges.
