Simplifying The Expression (x/4 + 7/8) / X^2 A Step-by-Step Guide

In mathematics, simplifying expressions is a fundamental skill. It allows us to represent mathematical relationships in their most concise and understandable form. One common type of expression that often requires simplification is a complex fraction. A complex fraction is a fraction where the numerator, the denominator, or both contain fractions themselves. These expressions can appear daunting at first, but with a systematic approach, they can be easily simplified. In this article, we will delve into the process of simplifying a complex fraction, providing a step-by-step guide that will equip you with the necessary tools to tackle similar problems. We will begin by understanding the given expression and identifying the key steps involved in its simplification. The ability to simplify complex fractions is crucial in various areas of mathematics, including algebra, calculus, and beyond. It enables us to solve equations, analyze functions, and gain a deeper understanding of mathematical concepts. By mastering this skill, you will enhance your mathematical proficiency and be well-prepared for more advanced topics. Throughout this guide, we will emphasize clarity and precision, ensuring that each step is explained thoroughly and logically. We will also provide examples to illustrate the concepts and techniques discussed. By the end of this article, you will have a solid understanding of how to simplify complex fractions and be able to apply this knowledge to a wide range of problems. Remember, practice is key to mastering any mathematical skill. Work through the examples provided and try simplifying other complex fractions on your own. With dedication and perseverance, you will become proficient in simplifying expressions and unlock new possibilities in your mathematical journey.

Breaking Down the Expression

To simplify the given expression, we must first understand its structure. The expression we are tasked with simplifying is:

$$\frac{\frac{x}{4}+\frac{7}{8}}{x^2}$$

This expression is a complex fraction because the numerator contains the sum of two fractions. Our goal is to combine these fractions in the numerator and then simplify the entire expression. The first step involves finding a common denominator for the fractions in the numerator. This will allow us to add the fractions together and consolidate the numerator into a single fraction. Once we have a single fraction in the numerator, we can proceed to divide this fraction by the denominator, which is $x^2$. Dividing by a fraction is equivalent to multiplying by its reciprocal, so we will use this principle to simplify the expression further. Throughout this process, we will pay close attention to algebraic manipulations and ensure that each step is performed correctly. We will also look for opportunities to cancel out common factors, which will help us to reduce the expression to its simplest form. This step-by-step approach will ensure that we arrive at the correct answer while maintaining clarity and accuracy. Understanding the structure of the expression is crucial for developing a strategy for simplification. By breaking down the expression into smaller, more manageable parts, we can tackle each part individually and then combine the results to obtain the final simplified form. This approach is applicable to a wide range of mathematical problems and is a valuable skill to develop. As we proceed through the simplification process, we will highlight key concepts and techniques that are essential for success. We will also provide explanations and justifications for each step, ensuring that you understand the reasoning behind the manipulations. This will not only help you to solve the given problem but also enhance your overall mathematical understanding and problem-solving abilities.

Finding the Common Denominator

The key to simplifying the numerator, $\frac{x}{4}+\frac{7}{8}$, lies in finding a common denominator. The denominators in this case are 4 and 8. The least common multiple (LCM) of 4 and 8 is 8. This means we need to rewrite the fraction $\frac{x}{4}$ so that it has a denominator of 8. To do this, we multiply both the numerator and the denominator of $\frac{x}{4}$ by 2:

$$\frac{x}{4} \times \frac{2}{2} = \frac{2x}{8}$$

Now we can rewrite the original expression in the numerator with the common denominator:

$$\frac{2x}{8} + \frac{7}{8}$$

With a common denominator, we can now add the fractions by adding their numerators:

$$\frac{2x + 7}{8}$$

This combined fraction represents the simplified form of the numerator. We have successfully consolidated the two fractions into a single fraction, which is a crucial step in simplifying the overall expression. The process of finding a common denominator is a fundamental skill in working with fractions. It allows us to perform addition and subtraction operations on fractions with different denominators. The least common multiple (LCM) is the smallest number that is a multiple of both denominators, and it serves as the ideal common denominator. By using the LCM, we minimize the need for further simplification later on. In this case, the LCM of 4 and 8 is 8, which made the conversion straightforward. However, in other cases, finding the LCM may require more effort, especially when dealing with larger numbers or algebraic expressions. There are various methods for finding the LCM, including listing multiples, prime factorization, and using the greatest common divisor (GCD). Mastering these methods will greatly enhance your ability to work with fractions and simplify expressions. Once we have found the common denominator, we can proceed to add or subtract the fractions as needed. The resulting fraction will have the common denominator, and the numerator will be the sum or difference of the original numerators. This process allows us to combine multiple fractions into a single fraction, which is often a necessary step in simplifying more complex expressions.

Dividing by $x^2$

Now that we've simplified the numerator to $\frac{2x + 7}{8}$, we can rewrite the original expression as:

$$\frac{\frac{2x + 7}{8}}{x^2}$$

Dividing by $x^2$ is the same as multiplying by its reciprocal, which is $\frac{1}{x^2}$. So, we can rewrite the expression as:

$$\frac{2x + 7}{8} \times \frac{1}{x^2}$$

To multiply fractions, we multiply the numerators together and the denominators together:

$$\frac{(2x + 7) \times 1}{8 \times x^2} = \frac{2x + 7}{8x^2}$$

This is the simplified form of the original expression. We have successfully combined the fractions in the numerator, divided by the denominator, and expressed the result as a single fraction. The process of dividing by a fraction or an expression involves multiplying by its reciprocal. This is a fundamental principle in algebra and is essential for simplifying complex fractions and other algebraic expressions. The reciprocal of a fraction is obtained by swapping the numerator and the denominator. For example, the reciprocal of $\frac{a}{b}$ is $\frac{b}{a}$. Similarly, the reciprocal of an expression like $x^2$ is $\frac{1}{x^2}$. When dividing by a fraction, we multiply by its reciprocal, which effectively inverts the fraction and changes the operation from division to multiplication. This allows us to combine the fractions more easily and simplify the expression. In this case, dividing by $x^2$ was equivalent to multiplying by $\frac{1}{x^2}$, which allowed us to combine the numerator and the denominator into a single fraction. The resulting fraction, $\frac{2x + 7}{8x^2}$, represents the simplified form of the original expression. This process highlights the importance of understanding the relationship between division and multiplication and how reciprocals can be used to simplify expressions.

Final Simplified Expression

The final simplified expression is:

$$\frac{2x + 7}{8x^2}$$

Comparing this result with the given options, we see that none of the options match this simplified form directly. However, let's analyze the options provided and see if any of them are algebraically equivalent to our result.

The given options are:

A) $\frac{x^3}{40}$ B) $\frac{1}{4}$ C) $\frac{64+x^3}{64 x}$

Our simplified expression, $\frac{2x + 7}{8x^2}$, is a rational expression where the numerator is a linear expression (2x + 7) and the denominator is a quadratic expression (8x^2). None of the given options have the same form. Option A, $\frac{x^3}{40}$, is a simple rational expression with a cubic term in the numerator and a constant denominator. Option B, $\frac{1}{4}$, is a constant value and does not involve any variables. Option C, $\frac{64+x^3}{64 x}$, is a rational expression with a cubic term in the numerator and a linear term in the denominator. None of these options are algebraically equivalent to our simplified expression. This indicates that there may be an error in the provided options or in the original question. It is important to double-check the original expression and the steps taken during simplification to ensure that no mistakes were made. In this case, we have carefully reviewed our steps and are confident that our simplified expression, $\frac{2x + 7}{8x^2}$, is correct. Therefore, the correct answer is not among the given options. This highlights the importance of critical thinking and not simply choosing the closest option without verifying its correctness. In mathematics, it is essential to be precise and accurate, and sometimes the correct answer may not be explicitly provided. In such cases, it is important to identify the discrepancy and communicate it appropriately.

In conclusion, we have successfully simplified the given complex fraction $\frac{\frac{x}{4}+\frac{7}{8}}{x^2}$ to $\frac{2x + 7}{8x^2}$. Throughout this process, we have emphasized the importance of understanding the structure of the expression, finding a common denominator, and applying the rules of fraction manipulation. We have also highlighted the significance of verifying the final result and comparing it with the given options. In this case, we found that none of the provided options matched our simplified expression, which underscores the importance of critical thinking and accuracy in mathematics. The ability to simplify complex fractions is a valuable skill that is applicable in various areas of mathematics and beyond. By mastering this skill, you will enhance your problem-solving abilities and gain a deeper understanding of mathematical concepts. Remember, practice is key to success. Work through various examples and apply the techniques discussed in this article to different types of complex fractions. With dedication and perseverance, you will become proficient in simplifying expressions and unlock new possibilities in your mathematical journey. This guide has provided a comprehensive step-by-step approach to simplifying complex fractions, and we encourage you to use this knowledge to tackle more challenging problems. As you continue to explore mathematics, you will encounter a wide range of expressions that require simplification, and the skills you have learned here will serve you well. Embrace the challenges and continue to learn and grow in your mathematical pursuits.