Simplifying (10x-15)/6 * 1/(8x-12) A Step-by-Step Guide
In the realm of mathematics, simplifying and manipulating algebraic expressions is a fundamental skill. This article delves into a specific example: the multiplication of two rational expressions, (10x-15)/6 * 1/(8x-12). We will break down the process step-by-step, covering key concepts like factoring, simplifying fractions, and applying the rules of multiplication. By the end of this guide, you'll have a solid understanding of how to approach similar problems and confidently simplify complex expressions.
Understanding Rational Expressions
Before we dive into the specific problem, let's clarify what rational expressions are. Rational expressions are essentially fractions where the numerator and denominator are polynomials. Polynomials, in turn, are algebraic expressions containing variables and coefficients, combined using addition, subtraction, and multiplication, with non-negative integer exponents. Examples of polynomials include x^2 + 3x - 2, 5x^4 - 1, and even simple expressions like 7x or 9. Therefore, expressions like (x+1)/(x-2), (3x^2 - 5)/(x+4), and our target expression, (10x-15)/6 * 1/(8x-12), all fall under the umbrella of rational expressions.
The ability to work with rational expressions is crucial in various areas of mathematics, including algebra, calculus, and beyond. They appear frequently in equations, functions, and modeling real-world phenomena. Mastering the techniques for simplifying, adding, subtracting, multiplying, and dividing rational expressions opens doors to solving a wider range of mathematical problems.
The initial step in dealing with complex rational expressions often involves simplification through factoring. Factoring is the process of breaking down a polynomial into its constituent factors – expressions that, when multiplied together, yield the original polynomial. This is a cornerstone of simplifying rational expressions because it allows us to identify common factors in the numerator and denominator, which can then be canceled out, leading to a more concise expression. For example, the expression x^2 - 4 can be factored into (x+2)(x-2). Recognizing these factors is key to simplifying rational expressions that contain this term.
Step 1: Factoring the Expressions
Our first task in simplifying (10x-15)/6 * 1/(8x-12) is to factor the expressions in the numerators and denominators. Factoring is like reverse distribution; we're looking for common factors that can be pulled out. Let's start with the first expression, (10x-15)/6. In the numerator, 10x - 15, we can see that both terms are divisible by 5. Factoring out the 5, we get 5(2x - 3). The denominator, 6, is already a constant and doesn't need further factoring. So, (10x-15)/6 becomes 5(2x-3)/6.
Now, let's move on to the second expression, 1/(8x-12). The numerator is simply 1, which is already in its simplest form. In the denominator, 8x - 12, both terms are divisible by 4. Factoring out the 4, we get 4(2x - 3). Thus, 1/(8x-12) becomes 1/[4(2x-3)].
By factoring, we've transformed our original expression into [5(2x-3)/6] * [1/4(2x-3)]. Notice the presence of the common factor (2x-3) in both the numerator and the denominator. This is a crucial observation, as it sets the stage for the next step: simplification.
Factoring is not just a mechanical process; it's about recognizing patterns and relationships within the expressions. In this case, spotting the common factor of 5 in 10x - 15 and the common factor of 4 in 8x - 12 was essential. Practice with various factoring techniques, such as factoring out the greatest common factor, factoring trinomials, and recognizing special patterns like the difference of squares, will significantly enhance your ability to simplify rational expressions efficiently.
Step 2: Simplifying the Expressions
Now that we've factored the expressions, we have [5(2x-3)/6] * [1/4(2x-3)]. The next step is to simplify by canceling out any common factors that appear in both the numerator and the denominator. This is a fundamental principle of fraction simplification: if a factor is present in both the top and bottom of a fraction (or in the numerators and denominators of multiplied fractions), it can be canceled out without changing the value of the expression.
In our case, we have the common factor (2x-3). This factor appears once in the numerator of the first fraction and once in the denominator of the second fraction. Therefore, we can cancel it out. This leaves us with [5/6] * [1/4]. The expression has now been significantly simplified.
It's important to note that we can only cancel out factors that are multiplied, not terms that are added or subtracted. For example, we could cancel (2x-3) because it was multiplied by 5 in the first numerator and by 4 in the second denominator. However, if we had an expression like (5 + (2x-3))/6, we couldn't simply cancel the (2x-3) term. This distinction is crucial to avoid errors in simplification.
Simplifying rational expressions is not just about making them look simpler; it's about making them easier to work with. A simplified expression is often easier to evaluate, differentiate, integrate, or use in further calculations. The ability to simplify effectively is a cornerstone of advanced mathematical techniques.
Step 3: Multiplying the Simplified Fractions
After simplifying, we're left with [5/6] * [1/4]. Multiplying fractions is straightforward: we multiply the numerators together and the denominators together. So, 5/6 multiplied by 1/4 is equal to (5 * 1) / (6 * 4).
Performing the multiplication, we get 5 / 24. This is our final simplified expression. There are no more common factors between the numerator (5) and the denominator (24), so the fraction is in its simplest form.
Therefore, the simplified form of the original expression, (10x-15)/6 * 1/(8x-12), is 5/24. This constant value highlights how simplifying rational expressions can sometimes lead to surprising results. An expression that initially appears complex can, through factoring and simplification, be reduced to a simple number.
The process of multiplying rational expressions involves more than just following rules; it's about understanding the underlying principles of fractions and algebraic manipulation. The ability to confidently multiply fractions, both numerical and algebraic, is a fundamental skill in mathematics and has wide-ranging applications in various fields.
Conclusion: Mastering Rational Expression Multiplication
In this article, we've walked through the process of simplifying and multiplying the rational expression (10x-15)/6 * 1/(8x-12). We began by understanding what rational expressions are and why they're important. Then, we tackled the problem step-by-step, focusing on three key phases: factoring, simplifying, and multiplying.
The first step, factoring, allowed us to identify common factors within the numerators and denominators. This is a crucial skill for simplifying any rational expression. The second step, simplifying, involved canceling out these common factors, reducing the expression to its simplest form. Finally, the third step, multiplying, involved multiplying the simplified fractions together to arrive at our final answer: 5/24.
This example demonstrates the power of algebraic manipulation. What started as a seemingly complex expression was ultimately reduced to a simple constant value. This highlights the importance of mastering the techniques of factoring, simplifying, and multiplying rational expressions. These skills are not only essential for success in algebra and calculus but also provide a foundation for tackling more advanced mathematical concepts.
The journey of simplifying rational expressions doesn't end here. There are numerous other examples and variations to explore, each presenting unique challenges and opportunities for learning. The more you practice these techniques, the more comfortable and confident you'll become in your ability to manipulate algebraic expressions. Remember, mathematics is a journey of continuous learning and discovery, and the ability to simplify and multiply rational expressions is a valuable step along the way.
Keep practicing, keep exploring, and keep simplifying!
