Simplifying Complex Rational Expressions A Step-by-Step Guide

In the realm of mathematics, complex rational expressions, also known as complex fractions, often present a challenge to students and enthusiasts alike. These expressions involve fractions within fractions, requiring a systematic approach to simplify them effectively. This article delves into the intricacies of simplifying a specific complex rational expression, providing a detailed step-by-step solution and highlighting the underlying principles involved. Understanding these principles is crucial not only for solving similar problems but also for building a stronger foundation in algebraic manipulation. The expression we will tackle is:

$$\frac{\frac{6}{x+3}+\frac{9}{x+6}}{\frac{5 x+21}{x^2+9 x+18}}$$

This expression appears daunting at first glance, but by breaking it down into smaller, manageable steps, we can navigate through the simplification process with clarity and precision. Our journey will involve finding common denominators, combining fractions, factoring polynomials, and ultimately, simplifying the expression to its most concise form. Throughout this process, we'll emphasize the importance of each step, ensuring a thorough understanding of the techniques employed. Mastering the art of simplifying complex fractions not only enhances problem-solving skills but also provides a deeper appreciation for the elegance and structure inherent in mathematical expressions.

Step 1: Finding a Common Denominator for the Numerator

To begin simplifying the complex fraction, our initial focus lies on the numerator, which comprises two individual fractions: $\frac{6}{x+3}$ and $\frac{9}{x+6}$. Before we can combine these fractions, we must first identify a common denominator. The common denominator is the least common multiple (LCM) of the denominators, which in this case are $(x+3)$ and $(x+6)$. Since these expressions do not share any common factors, their LCM is simply their product: $(x+3)(x+6)$. Now, we need to rewrite each fraction with this common denominator. To do this, we multiply the numerator and denominator of each fraction by the appropriate factor. For the first fraction, $\frac{6}{x+3}$, we multiply both the numerator and denominator by $(x+6)$: $\frac6}{x+3} \cdot \frac{x+6}{x+6} = \frac{6(x+6)}{(x+3)(x+6)} = \frac{6x+36}{(x+3)(x+6)}$ For the second fraction, $\frac{9}{x+6}$, we multiply both the numerator and denominator by $(x+3)$ $\frac{9{x+6} \cdot \frac{x+3}{x+3} = \frac{9(x+3)}{(x+3)(x+6)} = \frac{9x+27}{(x+3)(x+6)}$ By rewriting the fractions with a common denominator, we have set the stage for combining them, a crucial step in simplifying the overall complex fraction. This meticulous approach ensures that we maintain the equivalence of the expression while making it more amenable to further manipulation. The ability to identify and utilize common denominators is a fundamental skill in algebra, allowing us to perform operations on fractions with confidence and accuracy. This step not only simplifies the current expression but also lays the groundwork for more advanced algebraic techniques, such as solving rational equations and inequalities. By mastering this foundational concept, students can unlock a deeper understanding of mathematical relationships and tackle complex problems with greater ease.

Step 2: Combining the Fractions in the Numerator

With a common denominator of $(x+3)(x+6)$ now established, we can seamlessly combine the two fractions in the numerator. This involves adding the numerators while keeping the common denominator intact. Recall that we have the following equivalent fractions: $\frac6x+36}{(x+3)(x+6)} ext{ and } \frac{9x+27}{(x+3)(x+6)}$ Adding these fractions together, we get $\frac{6x+36(x+3)(x+6)} + \frac{9x+27}{(x+3)(x+6)} = \frac{(6x+36) + (9x+27)}{(x+3)(x+6)}$ Now, we simplify the numerator by combining like terms $\frac{6x+36+9x+27(x+3)(x+6)} = \frac{15x+63}{(x+3)(x+6)}$ This step demonstrates the fundamental principle of adding fractions once a common denominator is achieved, the numerators can be directly combined. The resulting fraction, $\frac{15x+63(x+3)(x+6)}$, represents a significant simplification of the original numerator. However, our work is not yet complete. We can further simplify this fraction by factoring out the greatest common factor (GCF) from the numerator. In this case, the GCF of $15x$ and $63$ is $3$. Factoring out the $3$, we get $\frac{3(5x+21){(x+3)(x+6)}$ This factored form not only simplifies the expression but also reveals potential cancellations with the denominator of the complex fraction, which we will explore in the subsequent steps. The ability to combine fractions and simplify the resulting expression is a cornerstone of algebraic manipulation. It allows us to transform complex expressions into more manageable forms, paving the way for further analysis and problem-solving. By mastering this skill, students gain a valuable tool for tackling a wide range of mathematical challenges, from solving equations to working with rational functions.

Step 3: Factoring the Denominator of the Main Fraction

Now, let's shift our attention to the denominator of the main complex fraction, which is $\frac{5x+21}{x^2+9x+18}$. To simplify the complex fraction, we'll need to factor the quadratic expression in the denominator, $x^2+9x+18$. Factoring a quadratic expression involves finding two binomials that, when multiplied together, yield the original quadratic. In this case, we are looking for two numbers that add up to $9$ (the coefficient of the $x$ term) and multiply to $18$ (the constant term). The numbers $3$ and $6$ satisfy these conditions, since $3 + 6 = 9$ and $3 \cdot 6 = 18$. Therefore, we can factor the quadratic expression as follows: $x^2+9x+18 = (x+3)(x+6)$ Now, we can rewrite the denominator of the main fraction in its factored form: $\frac{5x+21}{x^2+9x+18} = \frac{5x+21}{(x+3)(x+6)}$ This step highlights the importance of factoring in simplifying rational expressions. By factoring the quadratic expression, we expose common factors that may exist between the numerator and denominator, allowing for cancellation and further simplification. The ability to factor quadratic expressions is a fundamental skill in algebra, with applications extending beyond simplifying fractions to solving quadratic equations and analyzing polynomial functions. Mastering this skill requires a solid understanding of factoring techniques, such as identifying pairs of numbers that satisfy the required sum and product conditions. Furthermore, it reinforces the concept of reverse distribution, where we break down a polynomial into its constituent factors. By factoring the denominator, we have not only simplified the expression but also prepared it for the next crucial step: dividing fractions. This step sets the stage for the final simplification, where we will invert and multiply, ultimately revealing the simplified form of the complex fraction. The process of factoring is a powerful tool in the mathematician's arsenal, enabling us to unravel the structure of polynomials and unlock their hidden relationships.

Step 4: Dividing Fractions by Inverting and Multiplying

Now that we have simplified both the numerator and the denominator of the complex fraction, we are ready to perform the division. Recall that dividing by a fraction is equivalent to multiplying by its reciprocal. Our complex fraction, after the previous steps, looks like this: $\frac\frac{3(5x+21)}{(x+3)(x+6)}}{\frac{5x+21}{(x+3)(x+6)}}$ To divide, we invert the denominator and multiply it by the numerator $\frac{3(5x+21){(x+3)(x+6)} \div \frac{5x+21}{(x+3)(x+6)} = \frac{3(5x+21)}{(x+3)(x+6)} \cdot \frac{(x+3)(x+6)}{5x+21}$ This step is a crucial transformation, converting a division problem into a multiplication problem, which is often easier to handle. The act of inverting the denominator and multiplying is a fundamental rule in fraction arithmetic, ensuring that we maintain the equivalence of the expression while simplifying it. This rule stems from the definition of division as the inverse operation of multiplication. By inverting the denominator, we are essentially finding the multiplicative inverse, which, when multiplied by the original fraction, results in the same value as dividing by the original denominator. This concept is not only essential for simplifying complex fractions but also for solving equations involving fractions and understanding rational functions. The ability to confidently apply this rule demonstrates a strong grasp of fraction arithmetic and algebraic manipulation. Furthermore, it lays the groundwork for more advanced mathematical concepts, such as working with rational expressions and functions. By mastering the technique of dividing fractions by inverting and multiplying, students equip themselves with a powerful tool for tackling a wide range of mathematical challenges.

Step 5: Simplifying the Expression by Cancelling Common Factors

With the division transformed into multiplication, we now have: $\frac3(5x+21)}{(x+3)(x+6)} \cdot \frac{(x+3)(x+6)}{5x+21}$ The final step in simplifying the complex fraction involves canceling out common factors that appear in both the numerator and the denominator. This process relies on the principle that any factor divided by itself equals 1, effectively removing it from the expression. Examining the expression, we can identify the following common factors $(5x+21)$ and $(x+3)(x+6)$. Canceling these factors, we get: $\frac{3\cancel{(5x+21)}{\cancel{(x+3)(x+6)}} \cdot \frac{\cancel{(x+3)(x+6)}}{\cancel{5x+21}} = 3$ Thus, the simplified form of the complex fraction is simply $3$. This final simplification demonstrates the power of factoring and canceling common factors in reducing complex expressions to their simplest forms. It highlights the elegance and efficiency of mathematical operations, where seemingly intricate expressions can be distilled down to concise results. The ability to identify and cancel common factors is a crucial skill in algebra, allowing us to simplify rational expressions, solve equations, and analyze functions. This skill reinforces the understanding of factors, multiples, and the fundamental operations of arithmetic. Furthermore, it emphasizes the importance of paying attention to detail and carefully examining expressions for potential simplifications. By mastering the art of canceling common factors, students gain a valuable tool for tackling a wide range of mathematical challenges and developing a deeper appreciation for the structure and patterns inherent in mathematical expressions. In this case, the simplification to a constant value of 3 underscores the hidden simplicity within the original complex fraction, revealing the underlying mathematical relationship in its most basic form.

In conclusion, we have successfully simplified the complex rational expression $\frac{\frac{6}{x+3}+\frac{9}{x+6}}{\frac{5 x+21}{x^2+9 x+18}}$ to its simplest form, which is $3$. This process involved a series of key steps, each building upon the previous one: finding a common denominator, combining fractions, factoring polynomials, inverting and multiplying, and finally, canceling common factors. Each step underscores fundamental principles of algebra and fraction arithmetic. The journey through this simplification process not only provides a solution to the specific problem but also reinforces essential mathematical skills. These skills are crucial for tackling more complex problems in algebra, calculus, and beyond. The ability to manipulate fractions, factor polynomials, and simplify expressions is a cornerstone of mathematical proficiency. Furthermore, this exercise highlights the importance of a systematic and organized approach to problem-solving. By breaking down a complex problem into smaller, manageable steps, we can navigate through the challenges with greater clarity and confidence. The techniques used in this simplification are not limited to this specific problem; they are widely applicable to a variety of mathematical contexts. Mastering these techniques empowers students to approach new problems with a strategic mindset and a toolbox of effective methods. Ultimately, the simplification of complex rational expressions is not just about finding the answer; it's about developing a deeper understanding of mathematical principles and honing problem-solving skills that will serve as a foundation for future success in mathematics and related fields. The final result, the constant value of 3, is a testament to the hidden simplicity that often lies beneath the surface of complex mathematical expressions. This simplification serves as a reminder that even the most daunting problems can be solved through careful application of fundamental principles and a systematic approach.