Rewriting Rational Expressions With Common Denominator (a-5)(a-7)(a-8)

In algebra, rewriting rational expressions to have a common denominator is a fundamental skill. This process is essential for adding, subtracting, and comparing rational expressions. When dealing with fractions, a common denominator allows us to combine or compare numerators directly. In this article, we will walk through the process of rewriting two given rational expressions, 9/(a^2 - 12a + 35) and (5a)/(a^2 - 13a + 40), so they share the common denominator (a - 5)(a - 7)(a - 8). This technique is widely used in various mathematical contexts, including calculus and equation solving. Mastering this skill is crucial for simplifying complex algebraic expressions and solving equations involving rational functions. By the end of this article, you will understand the steps involved in finding the least common denominator and rewriting rational expressions effectively.

Before we dive into the process, let's define what a rational expression is. A rational expression is essentially a fraction where the numerator and denominator are polynomials. For example, (x + 1) / (x^2 - 4) is a rational expression. The key here is that the denominator cannot be zero, as division by zero is undefined. Therefore, when dealing with rational expressions, we often need to identify values that would make the denominator zero and exclude them from the domain. This is why factoring and simplifying rational expressions are essential steps in algebraic manipulations. In the context of the given problem, we have two rational expressions: 9/(a^2 - 12a + 35) and (5a)/(a^2 - 13a + 40). Our goal is to rewrite these expressions with a common denominator, which requires us to factor the denominators and identify the least common multiple of the factors. The process involves several algebraic techniques, including factoring quadratic expressions and multiplying rational expressions. By mastering these techniques, we can simplify complex expressions and solve various algebraic problems more efficiently. Understanding rational expressions and their properties is a cornerstone of algebra, and it is a skill that is frequently applied in higher-level mathematics.

The first step in rewriting our rational expressions is to factor the denominators. Factoring helps us identify the common and unique factors that will form our common denominator. Let's start with the first denominator: a^2 - 12a + 35. We need to find two numbers that multiply to 35 and add up to -12. These numbers are -5 and -7. So, we can factor the quadratic expression as (a - 5)(a - 7). Now, let's factor the second denominator: a^2 - 13a + 40. We need to find two numbers that multiply to 40 and add up to -13. These numbers are -5 and -8. Therefore, the quadratic expression can be factored as (a - 5)(a - 8). Factoring quadratic expressions is a critical skill in algebra, and it involves understanding the relationship between the coefficients and the roots of the quadratic. By breaking down the denominators into their factors, we can clearly see the components we need for our common denominator. In this case, we have the factors (a - 5), (a - 7), and (a - 8). The next step is to use these factors to construct the least common denominator, which will allow us to rewrite the rational expressions with a unified denominator. Factoring also helps in simplifying rational expressions by canceling out common factors between the numerator and the denominator, making the expressions easier to work with.

Factoring $a^2 - 12a + 35$

To factor the quadratic expression a^2 - 12a + 35, we are looking for two numbers that multiply to 35 and add to -12. The numbers -5 and -7 satisfy these conditions since (-5) * (-7) = 35 and (-5) + (-7) = -12. Therefore, we can rewrite the quadratic expression as (a - 5)(a - 7). This factorization is a fundamental step in simplifying and manipulating algebraic expressions. Factoring allows us to break down complex expressions into simpler components, making it easier to identify common factors and simplify fractions. In the context of rational expressions, factoring the denominator is crucial for finding the least common denominator (LCD), which is essential for adding, subtracting, and comparing fractions. The ability to factor quadratic expressions like a^2 - 12a + 35 is a cornerstone of algebra, and it is a skill that is frequently applied in various mathematical contexts, including calculus and equation solving. Mastering this skill enhances one's ability to handle more complex algebraic problems and perform operations with rational functions effectively. This specific factorization helps us in the broader context of rewriting rational expressions with a common denominator, which is a common task in algebraic manipulations and simplifications.

Factoring $a^2 - 13a + 40$

Factoring the quadratic expression a^2 - 13a + 40 involves finding two numbers that multiply to 40 and add up to -13. These numbers are -5 and -8, as (-5) * (-8) = 40 and (-5) + (-8) = -13. Thus, the factored form of the expression is (a - 5)(a - 8). This step is crucial for understanding the structure of the denominator and identifying common factors that will be necessary for finding the least common denominator (LCD). Factoring quadratic expressions is a fundamental skill in algebra and is widely used in various areas of mathematics, such as solving quadratic equations, simplifying rational expressions, and analyzing polynomial functions. The process of factoring helps in breaking down complex expressions into simpler, more manageable components, making algebraic manipulations more straightforward. In the case of rational expressions, factoring the denominator allows us to determine the factors that are needed to create a common denominator, which is essential for adding, subtracting, and comparing rational fractions. The factorization (a - 5)(a - 8) is a critical piece in our puzzle, as it reveals the factors that we will need to consider when rewriting the given rational expressions with a common denominator. The ability to factor quadratic expressions efficiently and accurately is a key asset in algebraic problem-solving.

Now that we've factored the denominators, we can identify the common denominator. The denominators are (a - 5)(a - 7) and (a - 5)(a - 8). The common denominator is the least common multiple (LCM) of these two expressions. To find the LCM, we take each unique factor and its highest power present in any of the denominators. In this case, the unique factors are (a - 5), (a - 7), and (a - 8). Each factor appears only once in the denominators, so we take each of them as they are. Thus, the common denominator is (a - 5)(a - 7)(a - 8). Identifying the common denominator is a crucial step in adding or subtracting fractions, including rational expressions. It ensures that we can combine the numerators correctly once the fractions have the same denominator. The least common multiple approach is particularly useful because it gives us the smallest possible common denominator, which simplifies the subsequent calculations. Understanding how to find the LCM is essential not only for algebra but also for various areas of mathematics where fraction manipulation is necessary. In our context, the common denominator (a - 5)(a - 7)(a - 8) sets the stage for rewriting the original rational expressions so that they can be combined or compared if needed.

With the common denominator identified as (a - 5)(a - 7)(a - 8), we can now rewrite each rational expression. To do this, we multiply the numerator and denominator of each expression by the factors needed to obtain the common denominator. For the first expression, 9/(a^2 - 12a + 35), which we factored as 9/((a - 5)(a - 7)), we need to multiply both the numerator and the denominator by (a - 8). This gives us 9(a - 8) / ((a - 5)(a - 7)(a - 8)). For the second expression, (5a)/(a^2 - 13a + 40), which we factored as (5a)/((a - 5)(a - 8)), we need to multiply both the numerator and the denominator by (a - 7). This gives us (5a)(a - 7) / ((a - 5)(a - 7)(a - 8)). Rewriting rational expressions with a common denominator is a fundamental skill in algebra. It allows us to perform operations such as addition and subtraction on rational expressions, which is a common task in many mathematical contexts. The key is to ensure that we multiply both the numerator and the denominator by the same factors, which preserves the value of the expression while changing its form. This process is akin to finding equivalent fractions, which is a concept familiar from basic arithmetic. By rewriting the given rational expressions with the common denominator, we have transformed them into a form where they can be easily combined or compared, making further algebraic manipulations more manageable. This step is crucial for solving equations involving rational expressions and for simplifying complex algebraic expressions.

Rewriting $\frac{9}{a^2-12 a+35}$

To rewrite the rational expression 9 / (a^2 - 12a + 35) with the common denominator (a - 5)(a - 7)(a - 8), we first recall that a^2 - 12a + 35 factors to (a - 5)(a - 7). This means our expression is currently in the form 9 / [(a - 5)(a - 7)]. To achieve the desired common denominator, we need to multiply both the numerator and the denominator by the missing factor, which is (a - 8). Thus, we multiply the numerator 9 by (a - 8) and the denominator (a - 5)(a - 7) by (a - 8). This results in the expression [9(a - 8)] / [(a - 5)(a - 7)(a - 8)]. This process ensures that we are only changing the form of the rational expression, not its value, as multiplying both the numerator and the denominator by the same factor is equivalent to multiplying by 1. The rewritten expression is now in a form that shares a common denominator with other rational expressions, making it possible to perform operations such as addition and subtraction. This is a crucial step in simplifying and solving algebraic problems involving rational expressions. The ability to rewrite rational expressions with a common denominator is a fundamental skill in algebra, and it is frequently used in higher-level mathematics, such as calculus and differential equations. The specific step of multiplying by (a - 8) completes the transformation of the first rational expression to the desired form.

Rewriting $\frac{5 a}{a^2-13 a+40}$

To rewrite the rational expression (5a) / (a^2 - 13a + 40) with the common denominator (a - 5)(a - 7)(a - 8), we begin by recalling that a^2 - 13a + 40 factors to (a - 5)(a - 8). Therefore, the expression is currently in the form (5a) / [(a - 5)(a - 8)]. To obtain the desired common denominator, we need to multiply both the numerator and the denominator by the missing factor, which is (a - 7). This involves multiplying the numerator 5a by (a - 7) and the denominator (a - 5)(a - 8) by (a - 7). This results in the expression [5a(a - 7)] / [(a - 5)(a - 7)(a - 8)]. This process ensures that we are creating an equivalent rational expression, as multiplying both the numerator and the denominator by the same factor does not change the value of the expression. The rewritten expression now has the common denominator (a - 5)(a - 7)(a - 8), which allows us to perform operations such as addition and subtraction with other rational expressions that share this denominator. Rewriting rational expressions with a common denominator is a fundamental technique in algebra, crucial for simplifying complex expressions and solving equations involving rational functions. The ability to perform this operation accurately and efficiently is a key skill in algebraic manipulation and is essential for success in higher-level mathematics. The specific step of multiplying by (a - 7) completes the transformation of the second rational expression to the desired form.

After rewriting both rational expressions, we have:

• 9(a - 8) / ((a - 5)(a - 7)(a - 8))
• (5a)(a - 7) / ((a - 5)(a - 7)(a - 8))

These expressions are now equivalent to the original expressions but have the common denominator (a - 5)(a - 7)(a - 8). This form allows us to perform further operations, such as addition or subtraction, more easily. The process of rewriting rational expressions with a common denominator is a crucial step in various algebraic manipulations. It simplifies the process of combining or comparing rational expressions, which is often necessary in solving equations, simplifying complex fractions, and performing calculus operations. The final result, with both expressions sharing the same denominator, demonstrates the successful application of this technique. These expressions can now be combined or manipulated further as needed, making this a valuable skill in the toolbox of any algebra student. Understanding and mastering this process is essential for more advanced topics in mathematics, where rational expressions frequently appear in various contexts.

In conclusion, rewriting rational expressions with a common denominator is a critical skill in algebra. We began with the expressions 9/(a^2 - 12a + 35) and (5a)/(a^2 - 13a + 40) and successfully rewrote them with the common denominator (a - 5)(a - 7)(a - 8). This process involved factoring the denominators, identifying the least common multiple, and multiplying the numerators and denominators by the appropriate factors. This technique is fundamental for performing operations such as addition and subtraction on rational expressions. Mastering this skill allows for simplification of complex algebraic expressions and is a crucial step in solving equations involving rational functions. The ability to manipulate rational expressions effectively is essential for success in higher-level mathematics, including calculus and differential equations. By understanding the steps involved in finding a common denominator and rewriting expressions, students can confidently tackle more complex algebraic problems. The techniques discussed in this article provide a solid foundation for further exploration in algebra and related fields. This skill is not only valuable in academic settings but also in practical applications where algebraic manipulation is necessary. Therefore, understanding and practicing this process is highly beneficial for anyone studying mathematics or related disciplines.