Simplifying (5c)/(a^2 + Ab) - C/(a + B) A Step-by-Step Guide

Jul 15, 2025 by ADMIN 61 views

$Simplifying the Expression: \\frac{5c}{a^2 + ab} - \\frac{c}{a + b}$

In mathematics, simplifying expressions is a fundamental skill, particularly when dealing with algebraic fractions. This article delves into the process of simplifying a specific rational expression: $\frac{5c}{a^2 + ab} - \frac{c}{a + b}$ . We'll break down the steps involved, ensuring a clear understanding of the underlying principles. This process involves finding common denominators, combining fractions, and simplifying the resulting expression. Mastery of these techniques is crucial for success in algebra and calculus. Understanding how to manipulate algebraic fractions is not only essential for simplifying expressions but also for solving equations and tackling more complex mathematical problems.

Understanding the Initial Expression

To begin simplifying, let's first examine the expression we're working with: $\frac{5c}{a^2 + ab} - \frac{c}{a + b}$ . This expression involves two fractions, each containing algebraic terms. The first fraction has a numerator of $5c$ and a denominator of $a^2 + ab$ . The second fraction has a numerator of $c$ and a denominator of $a + b$ . Our goal is to combine these two fractions into a single, simplified fraction. Before we can combine them, we need to find a common denominator. Looking at the denominators, $a^2 + ab$ and $a + b$ , we can see that they share a common factor. The denominator $a^2 + ab$ can be factored, which will help us identify the least common denominator (LCD). Factoring is a critical step in simplifying rational expressions, as it allows us to identify common factors and simplify the expression more easily. By understanding the structure of the initial expression, we can strategically approach the simplification process. This involves recognizing the components of each fraction and how they relate to each other. A clear understanding of the initial expression is the foundation for successful simplification.

Factoring the Denominators

The next step in simplifying the expression $\frac{5c}{a^2 + ab} - \frac{c}{a + b}$ is to factor the denominators. Factoring is a crucial technique in simplifying algebraic expressions, as it allows us to identify common factors and find the least common denominator (LCD). The denominator of the first fraction, $a^2 + ab$ , can be factored by taking out the common factor of $a$ . This gives us $a(a + b)$ . The denominator of the second fraction, $a + b$ , is already in its simplest form and cannot be factored further. Now that we've factored the denominators, we can easily identify the LCD. The LCD is the smallest expression that is divisible by both denominators. In this case, the LCD is $a(a + b)$ . This is because it includes both factors present in the denominators: $a$ and $(a + b)$ . Factoring the denominators is a vital step in the simplification process. It allows us to rewrite the fractions with a common denominator, which is essential for combining them. By identifying the common factors, we can simplify the expression and make it easier to work with. Accurate factoring is key to successfully simplifying rational expressions. The ability to factor algebraic expressions is a fundamental skill in algebra and is essential for simplifying complex expressions.

Finding the Least Common Denominator (LCD)

Having factored the denominators of our expression $\frac{5c}{a^2 + ab} - \frac{c}{a + b}$ , we now need to determine the least common denominator (LCD). The LCD is the smallest expression that is divisible by both denominators. In this case, the factored form of the first denominator is $a(a + b)$ , and the second denominator is $a + b$ . To find the LCD, we need to consider all the factors present in both denominators. The factors are $a$ and $(a + b)$ . The LCD must include each factor the greatest number of times it appears in any of the denominators. Here, $a$ appears once in the first denominator and not at all in the second. The factor $(a + b)$ appears once in both denominators. Therefore, the LCD is the product of these factors: $a(a + b)$ . Understanding how to find the LCD is crucial for combining fractions with different denominators. It allows us to rewrite the fractions with a common denominator, making it possible to add or subtract them. The LCD is the foundation for simplifying rational expressions. By correctly identifying the LCD, we can proceed with the next steps of the simplification process. The concept of the LCD is not only applicable to algebraic expressions but also to numerical fractions, highlighting its importance in mathematics.

Rewriting Fractions with the LCD

With the least common denominator (LCD) identified as $a(a + b)$ for the expression $\frac{5c}{a^2 + ab} - \frac{c}{a + b}$ , our next step is to rewrite each fraction with this LCD. This involves adjusting the numerators of the fractions so that the overall value of the fraction remains unchanged. The first fraction, $\frac{5c}{a^2 + ab}$ , already has the LCD as its denominator since $a^2 + ab = a(a + b)$ . Therefore, we don't need to modify this fraction. The second fraction, $\frac{c}{a + b}$ , needs to be adjusted. To get the LCD of $a(a + b)$ , we need to multiply the denominator $(a + b)$ by $a$ . To maintain the value of the fraction, we must also multiply the numerator $c$ by the same factor, $a$ . This gives us $\frac{c \cdot a}{(a + b) \cdot a} = \frac{ac}{a(a + b)}$ . Now, both fractions have the same denominator, the LCD, which allows us to combine them. Rewriting fractions with a common denominator is a fundamental step in adding or subtracting fractions. It ensures that we are working with comparable quantities. This process involves identifying the missing factors in each denominator and multiplying both the numerator and denominator by those factors. Accurate manipulation of fractions is crucial for simplifying expressions and solving equations in algebra and calculus. The ability to rewrite fractions with a common denominator is a key skill in mathematical problem-solving.

Combining the Fractions

Now that we have rewritten both fractions in the expression $\frac{5c}{a^2 + ab} - \frac{c}{a + b}$ with the least common denominator (LCD) of $a(a + b)$ , we can combine them. The first fraction remains $\frac{5c}{a(a + b)}$ , and the second fraction has been rewritten as $\frac{ac}{a(a + b)}$ . To combine the fractions, we subtract the numerators while keeping the denominator the same. This gives us: $\frac{5c - ac}{a(a + b)}$ . In this step, we are essentially performing the subtraction operation on the two fractions. By having a common denominator, we can directly subtract the numerators. The result is a single fraction with the LCD as the denominator and the difference of the numerators as the new numerator. Combining fractions is a crucial step in simplifying expressions. It allows us to reduce multiple fractions into a single fraction, which is often easier to work with. This process involves adding or subtracting the numerators while keeping the denominator constant. Accurate combination of fractions is essential for solving equations and simplifying complex expressions. Understanding the rules of fraction arithmetic is fundamental to success in algebra and beyond. This step sets the stage for further simplification of the expression by examining the numerator for potential factoring or cancellation of terms.

Simplifying the Numerator

Having combined the fractions in our expression $\frac{5c}{a^2 + ab} - \frac{c}{a + b}$ , we now have $\frac{5c - ac}{a(a + b)}$ . The next step is to simplify the numerator, $5c - ac$ . We can see that both terms in the numerator have a common factor of $c$ . By factoring out $c$ , we get $c(5 - a)$ . This factorization allows us to rewrite the fraction as $\frac{c(5 - a)}{a(a + b)}$ . Simplifying the numerator is an important step in reducing a fraction to its simplest form. It involves identifying and factoring out common factors, which can then be potentially canceled with factors in the denominator. Factoring is a fundamental technique in algebra and is used extensively in simplifying expressions and solving equations. By simplifying the numerator, we make the fraction easier to work with and identify potential cancellations. The ability to factor and simplify algebraic expressions is a key skill in mathematics. This step often reveals opportunities to further reduce the fraction by canceling common factors between the numerator and denominator. Careful attention to detail in factoring and simplifying the numerator is crucial for obtaining the correct simplified expression. This step prepares the expression for the final stage of simplification, which involves canceling common factors.

Canceling Common Factors

After simplifying the numerator of our expression $\frac{5c}{a^2 + ab} - \frac{c}{a + b}$ , we have the fraction $\frac{c(5 - a)}{a(a + b)}$ . Now, we look for any common factors between the numerator and the denominator that can be canceled. In this case, there are no common factors to cancel. The numerator has factors of $c$ and $(5 - a)$ , while the denominator has factors of $a$ and $(a + b)$ . Since there are no matching factors, the fraction is already in its simplest form. Canceling common factors is a crucial step in simplifying fractions. It involves identifying factors that appear in both the numerator and the denominator and dividing both by those factors. This process reduces the fraction to its lowest terms. When there are no common factors to cancel, as in this case, it means that the fraction is already in its simplest form. Recognizing when a fraction is fully simplified is an important part of the simplification process. This step ensures that we have reduced the expression as much as possible. The absence of common factors indicates that we have reached the final simplified form of the expression. In this particular example, the fraction $\frac{c(5 - a)}{a(a + b)}$ represents the simplified form of the original expression.

Final Simplified Expression

Having gone through the steps of factoring, finding the least common denominator, rewriting fractions, combining them, simplifying the numerator, and canceling common factors, we arrive at the final simplified expression for $\frac{5c}{a^2 + ab} - \frac{c}{a + b}$ . The simplified expression is $\frac{c(5 - a)}{a(a + b)}$ . This expression represents the original expression in its most reduced form. There are no further simplifications possible, as there are no common factors between the numerator and the denominator. The final simplified expression is the result of a systematic process of algebraic manipulation. It demonstrates the power of factoring, finding common denominators, and canceling common factors in simplifying complex expressions. This simplified form is easier to work with and provides a clearer representation of the relationship between the variables. The ability to simplify algebraic expressions is a fundamental skill in mathematics and is essential for solving equations, simplifying functions, and tackling more advanced mathematical problems. The final simplified expression is the culmination of our efforts and represents the most concise form of the original expression. This result underscores the importance of mastering algebraic techniques for effective problem-solving.

By following these steps, we have successfully simplified the given rational expression. Understanding these techniques is crucial for anyone studying algebra and beyond.