Simplifying Rational Expressions A Step-by-Step Guide

In the realm of mathematics, simplifying complex expressions is a fundamental skill. This article delves into the simplification of a specific rational expression: ${\frac{\frac{x+8}{x^2-49}}{\frac{x^2-64}{x-7}}}$. We will dissect the expression, breaking it down into manageable steps, and utilize key algebraic techniques to arrive at its simplest form. This exploration will not only enhance your understanding of rational expressions but also equip you with the tools to tackle similar problems with confidence.

Understanding Rational Expressions

Before we dive into the specifics of the given expression, it's crucial to grasp the basics of rational expressions. A rational expression is essentially a fraction where the numerator and denominator are polynomials. Polynomials, in turn, are expressions consisting of variables and coefficients, combined using addition, subtraction, and non-negative integer exponents. Examples of polynomials include x^2 + 3x - 2, 5x^4 - 7, and even simple terms like 8x or -12. When these polynomials form the numerator and denominator of a fraction, we have a rational expression. The expression we aim to simplify, ${\frac{\frac{x+8}{x^2-49}}{\frac{x^2-64}{x-7}}}$, fits this definition perfectly. It's a fraction where both the numerator and the denominator are themselves rational expressions. To effectively simplify such expressions, we need to be comfortable with factoring polynomials, identifying common factors, and understanding the rules of fraction division. These skills will allow us to manipulate the expression strategically, canceling out common terms and ultimately arriving at the most concise representation. Remember, simplifying rational expressions is not just about finding the right answer; it's about developing a deeper understanding of the underlying algebraic principles and honing your problem-solving abilities.

Step 1: Rewriting the Complex Fraction

The initial form of the expression, ${\frac{\frac{x+8}{x^2-49}}{\frac{x^2-64}{x-7}}}$, presents a complex fraction – a fraction within a fraction. To simplify this, we must first rewrite it as a division problem. Remember, a fraction bar signifies division. Therefore, the given expression can be interpreted as the rational expression (x+8)/(x^2-49) divided by the rational expression (x^2-64)/(x-7). To divide fractions, we employ a fundamental rule: we multiply by the reciprocal of the divisor. In simpler terms, we flip the second fraction (the divisor) and change the division operation to multiplication. Applying this to our expression, we get: ${\frac{x+8}{x^2-49} \div \frac{x^2-64}{x-7} = \frac{x+8}{x^2-49} \times \frac{x-7}{x^2-64}}$. This transformation is a crucial first step because it converts the complex fraction into a more manageable form – a product of two simpler rational expressions. Now, we can focus on simplifying each of these individual expressions and then multiplying them together. This approach breaks down the problem into smaller, more digestible parts, making the simplification process more systematic and less daunting. By rewriting the complex fraction as a multiplication problem, we've set the stage for the next critical step: factoring the polynomials.

Step 2: Factoring Polynomials

Factoring polynomials is a cornerstone of simplifying rational expressions. It allows us to identify common factors that can be canceled out, leading to a more concise expression. In our case, we have four polynomials that need factoring: x^2 - 49, x^2 - 64, x + 8, and x - 7. The first two, x^2 - 49 and x^2 - 64, are classic examples of the difference of squares pattern. This pattern states that a^2 - b^2 can be factored into (a + b)(a - b). Applying this to x^2 - 49, we recognize that it's x^2 - 7^2, so it factors into (x + 7)(x - 7). Similarly, x^2 - 64 is x^2 - 8^2, factoring into (x + 8)(x - 8). The other two polynomials, x + 8 and x - 7, are already in their simplest forms and cannot be factored further. Now, let's rewrite our expression with the factored polynomials: ${\frac{x+8}{x^2-49} \times \frac{x-7}{x^2-64} = \frac{x+8}{(x+7)(x-7)} \times \frac{x-7}{(x+8)(x-8)}}$. This factored form is crucial because it clearly reveals the common factors present in the numerator and the denominator. These common factors are the key to simplifying the expression further. By mastering factoring techniques, such as the difference of squares, we gain the ability to dissect complex polynomials into their fundamental components, making simplification a much more straightforward process. In the next step, we will leverage these factored forms to cancel out common factors and reduce the expression to its simplest terms.

Step 3: Canceling Common Factors

With the polynomials factored, we can now identify and cancel common factors between the numerator and the denominator. Our expression, in its factored form, is: ${\frac{x+8}{(x+7)(x-7)} \times \frac{x-7}{(x+8)(x-8)}}$. We can clearly see that the factor (x + 8) appears in both the numerator and the denominator. Similarly, the factor (x - 7) also appears in both. These common factors can be canceled out because any non-zero expression divided by itself equals 1. So, we cancel (x + 8) from the numerator of the first fraction and the denominator of the second fraction. We also cancel (x - 7) from the denominator of the first fraction and the numerator of the second fraction. After canceling these common factors, our expression becomes: ${\frac{\cancel{x+8}}{(x+7)\cancel{(x-7)}} \times \frac{\cancel{x-7}}{\cancel{(x+8)}(x-8)} = \frac{1}{x+7} \times \frac{1}{x-8}}$. This step highlights the power of factoring in simplifying rational expressions. By breaking down the polynomials into their factors, we expose the common elements that can be eliminated, leading to a more simplified form. It's important to remember that we can only cancel factors that are multiplied, not terms that are added or subtracted. Canceling common factors is a fundamental technique that streamlines the expression and brings us closer to the final simplified form. In the next step, we will multiply the remaining fractions to obtain the ultimate simplified rational expression.

Step 4: Multiplying Remaining Fractions

After canceling the common factors, we are left with the simplified fractions: ${\frac{1}{x+7} \times \frac{1}{x-8}}$. To complete the simplification, we need to multiply these remaining fractions. The rule for multiplying fractions is straightforward: multiply the numerators together and multiply the denominators together. In this case, multiplying the numerators (1 and 1) gives us 1. Multiplying the denominators (x + 7 and x - 8) gives us (x + 7)(x - 8). Therefore, our expression becomes: ${\frac{1}{x+7} \times \frac{1}{x-8} = \frac{1}{(x+7)(x-8)}}$. While this is a perfectly valid simplified form, it's often considered good practice to expand the denominator to provide a more complete representation. To expand (x + 7)(x - 8), we use the distributive property (often remembered by the acronym FOIL – First, Outer, Inner, Last). Multiplying the terms, we get: (x + 7)(x - 8) = x^2 - 8x + 7x - 56 = x^2 - x - 56. Thus, our final simplified rational expression is: ${\frac{1}{x^2 - x - 56}}$. This step demonstrates the final act of simplification – combining the remaining fractions into a single, concise expression. By multiplying the numerators and denominators, and then expanding the result (if necessary), we arrive at the ultimate simplified form of the original complex rational expression. The ability to multiply fractions and expand polynomial products is an essential skill in algebraic manipulation, allowing us to express mathematical expressions in their most elegant and efficient forms. In the conclusion, we will summarize the entire simplification process and highlight the key techniques employed.

Final Answer

In conclusion, simplifying the complex rational expression ${\frac{\frac{x+8}{x^2-49}}{\frac{x^2-64}{x-7}}}$ involves a series of crucial steps. First, we rewrote the complex fraction as a division problem and then transformed it into a multiplication problem by inverting the second fraction. This initial step set the stage for the subsequent simplifications. Next, we factored the polynomials, recognizing patterns like the difference of squares, to break them down into their fundamental components. Factoring allowed us to identify common factors that could be canceled out. Canceling these common factors from both the numerator and the denominator significantly simplified the expression. Finally, we multiplied the remaining fractions, combining the numerators and the denominators, and expanded the denominator to obtain the ultimate simplified form. Through these steps, we have demonstrated a systematic approach to simplifying rational expressions, emphasizing the importance of understanding fundamental algebraic principles and techniques. The final simplified form of the given expression is ${\frac{1}{x^2 - x - 56}}$. This result showcases the power of algebraic manipulation in transforming complex expressions into more manageable and understandable forms. By mastering these techniques, you can confidently tackle a wide range of mathematical problems involving rational expressions and beyond.