Step-by-Step Guide Simplifying Complex Rational Expression

In the realm of mathematics, simplifying complex expressions is a fundamental skill. This article delves into the process of simplifying a specific rational expression, offering a comprehensive guide suitable for students and enthusiasts alike. We will break down each step, ensuring a clear understanding of the underlying principles. The expression we aim to simplify is: $\frac{-\frac{1}{x-2}+\frac{4}{x-6}}{-\frac{8}{x-6}}$. Let's embark on this mathematical journey together!

Understanding Rational Expressions

Before diving into the simplification process, it's crucial to grasp the essence of rational expressions. Rational expressions are essentially fractions where the numerator and denominator are polynomials. Polynomials, in turn, are expressions involving variables and coefficients, combined using addition, subtraction, and non-negative integer exponents. Examples of polynomials include $x^2 + 3x - 2$, $5x^4 - 1$, and even simple expressions like $7x$ or just the number $9$. Recognizing these building blocks is the first step toward mastering rational expression simplification. The key takeaway is that a rational expression looks like one polynomial divided by another, and our goal is often to make these expressions look simpler and easier to work with. This might involve combining terms, canceling factors, or rewriting the expression in a more manageable form. The process often mirrors the way we simplify regular numerical fractions, but with the added complexity of variables and polynomials. Understanding this foundation makes the subsequent steps in simplifying the given expression much more intuitive and less daunting.

Step 1: Simplifying the Numerator

The heart of simplifying this complex fraction lies in tackling the numerator first. Our numerator is $-\frac{1}{x-2}+\frac{4}{x-6}$, which is a sum of two fractions. To combine these fractions, we need a common denominator. The least common denominator (LCD) for $(x-2)$ and $(x-6)$ is simply their product, $(x-2)(x-6)$. This is because both expressions are linear and share no common factors. With the common denominator identified, we proceed to rewrite each fraction with this new denominator. The first fraction, $-\frac{1}{x-2}$, is multiplied by $\frac{x-6}{x-6}$, resulting in $-\frac{1(x-6)}{(x-2)(x-6)}$. Similarly, the second fraction, $\frac{4}{x-6}$, is multiplied by $\frac{x-2}{x-2}$, yielding $\frac{4(x-2)}{(x-2)(x-6)}$. Now, both fractions share the same denominator, allowing us to combine them. We add the numerators: $-(x-6) + 4(x-2)$. Expanding this gives us $-x + 6 + 4x - 8$. Combining like terms, we arrive at $3x - 2$. Thus, the simplified numerator becomes $\frac{3x-2}{(x-2)(x-6)}$. This step is crucial as it transforms a sum of fractions into a single, manageable fraction, paving the way for further simplification. Remember, the goal is to make the expression as concise and understandable as possible, and simplifying the numerator is a significant stride in that direction.

Step 2: Rewriting the Main Expression

Having simplified the numerator to $\frac{3x-2}{(x-2)(x-6)}$, we can now rewrite the original expression. The expression $\frac{-\frac{1}{x-2}+\frac{4}{x-6}}{-\frac{8}{x-6}}$ transforms into $\frac{\frac{3x-2}{(x-2)(x-6)}}{-\frac{8}{x-6}}$. This rewriting is more than just a cosmetic change; it sets the stage for the next crucial step: dividing fractions. Remember that dividing by a fraction is the same as multiplying by its reciprocal. This is a fundamental rule of fraction manipulation, and it's the key to simplifying our complex expression. By recognizing this, we can convert the division problem into a multiplication problem, which is often easier to handle. The complex fraction, which might seem intimidating at first glance, is now on the verge of being simplified into a more manageable form. The act of rewriting allows us to clearly see the two fractions involved and apply the rule of reciprocals, bringing us closer to the final simplified expression. This step highlights the power of strategic manipulation in mathematics – by changing the form of the expression, we can make it easier to solve.

Step 3: Dividing Fractions (Multiplying by the Reciprocal)

The cornerstone of simplifying our expression lies in understanding how to divide fractions. As previously mentioned, dividing by a fraction is mathematically equivalent to multiplying by its reciprocal. The reciprocal of a fraction is simply flipping the numerator and the denominator. So, the reciprocal of $-\frac{8}{x-6}$ is $-\frac{x-6}{8}$. Now, we can rewrite our expression $\frac{\frac{3x-2}{(x-2)(x-6)}}{-\frac{8}{x-6}}$ as a multiplication problem: $\frac{3x-2}{(x-2)(x-6)} \times -\frac{x-6}{8}$. This transformation is a pivotal step because multiplication is often easier to handle than division, especially when dealing with complex fractions. By converting the division into multiplication, we've set the stage for potential cancellations and further simplification. This step exemplifies a common strategy in mathematics: transforming a problem into an equivalent form that is easier to solve. The ability to recognize and apply this principle is crucial for mastering not just rational expressions, but a wide range of mathematical concepts. With the expression now framed as a multiplication problem, we are well-positioned to identify common factors and simplify the expression to its most basic form.

Step 4: Canceling Common Factors

The beauty of rewriting our expression as a multiplication problem, $\frac{3x-2}{(x-2)(x-6)} \times -\frac{x-6}{8}$, now becomes evident. We can now look for common factors in the numerator and the denominator that can be canceled out. Notice that the term $(x-6)$ appears in both the numerator and the denominator. This common factor can be canceled, simplifying the expression significantly. Canceling $(x-6)$ from both the numerator and denominator, we are left with $\frac{3x-2}{(x-2)} \times -\frac{1}{8}$. This step is a testament to the power of simplification through factorization and cancellation. By identifying and eliminating common factors, we reduce the complexity of the expression and bring it closer to its simplest form. This process not only makes the expression easier to work with but also reveals the underlying structure and relationships between the different parts of the expression. The ability to spot and cancel common factors is a crucial skill in algebra and beyond. It allows us to streamline calculations, solve equations more efficiently, and gain a deeper understanding of mathematical relationships. With the $(x-6)$ term canceled, our expression is now significantly cleaner and ready for the final step of simplification.

Step 5: Final Simplification

After canceling the common factor, our expression stands as $\frac{3x-2}{(x-2)} \times -\frac{1}{8}$. Now, we simply multiply the remaining fractions together. Multiplying the numerators gives us $-(3x-2)$, which can be written as $-3x + 2$. Multiplying the denominators gives us $8(x-2)$. Therefore, the simplified expression is $\frac{-3x+2}{8(x-2)}$. While this form is perfectly acceptable, it's often considered good practice to leave the denominator in factored form. This makes it easier to identify any restrictions on the variable $x$ (in this case, $x$ cannot be 2, as it would make the denominator zero). So, our final simplified expression is $\frac{-3x+2}{8(x-2)}$. This completes the simplification process. We started with a complex fraction and, through a series of logical steps – finding common denominators, rewriting division as multiplication, canceling common factors, and multiplying – we arrived at a much simpler form. This process illustrates the elegance and power of algebraic manipulation. The final simplified expression is not only more concise but also easier to understand and work with in further calculations or applications. This journey through simplification highlights the importance of mastering fundamental algebraic techniques and the rewards of persistent, step-by-step problem-solving.

Conclusion

Simplifying rational expressions, like the one we tackled, might seem daunting initially, but by breaking down the process into manageable steps, it becomes quite approachable. We began by understanding the fundamental principles of rational expressions, progressed through simplifying the numerator, rewriting the main expression, dividing fractions by multiplying by the reciprocal, canceling common factors, and finally, arriving at the fully simplified form: $\frac{-3x+2}{8(x-2)}$. This journey underscores the importance of a systematic approach to problem-solving in mathematics. Each step built upon the previous one, ultimately leading to a clear and concise solution. Moreover, this exercise reinforces key algebraic skills such as finding common denominators, working with reciprocals, and identifying and canceling common factors. These skills are not only essential for simplifying rational expressions but also for tackling a wide range of mathematical problems. The ability to break down complex problems into smaller, more manageable steps is a valuable asset in mathematics and beyond. By mastering these techniques, students and enthusiasts can confidently approach even the most challenging mathematical expressions and unlock their hidden simplicity.