Simplify The Expression (5k/6) * (3/(2k^3)).

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In the realm of mathematics, particularly algebra, simplifying expressions is a fundamental skill. This article delves into the process of simplifying rational expressions, using a specific example to illustrate the steps involved. Our goal is to provide a clear, concise, and comprehensive guide that will empower you to tackle similar problems with confidence. Let's embark on this mathematical journey together, unraveling the intricacies of rational expressions and mastering the art of simplification.

Understanding the Basics of Rational Expressions

Before we dive into the simplification process, it's crucial to grasp the essence of rational expressions. A rational expression is simply a fraction where the numerator and the denominator are polynomials. Think of it as a ratio of two polynomial expressions. These expressions can involve variables, constants, and mathematical operations like addition, subtraction, multiplication, and division. Simplifying these expressions is akin to reducing a fraction to its lowest terms – we aim to eliminate common factors and arrive at the most concise form. In essence, we are looking for equivalent expressions that are easier to work with. This process often involves factoring polynomials, canceling common factors, and applying the rules of exponents. A solid understanding of these underlying principles is paramount to successfully navigating the world of rational expressions.

Key Concepts and Definitions

  • Polynomial: An expression consisting of variables and coefficients, involving only the operations of addition, subtraction, multiplication, and non-negative integer exponents.
  • Rational Expression: A fraction where both the numerator and denominator are polynomials.
  • Simplifying: The process of reducing a rational expression to its lowest terms by canceling common factors.
  • Factoring: The process of expressing a polynomial as a product of simpler polynomials.
  • Common Factors: Factors that appear in both the numerator and denominator of a rational expression.

Why Simplify Rational Expressions?

Simplifying rational expressions is not merely an academic exercise; it's a crucial skill with practical applications. Simplified expressions are easier to understand, manipulate, and use in further calculations. They can streamline complex equations, making them more manageable and less prone to errors. In various fields like engineering, physics, and computer science, simplified rational expressions play a vital role in modeling real-world phenomena and solving complex problems. By mastering the art of simplification, you equip yourself with a powerful tool that unlocks new possibilities and enhances your problem-solving prowess.

Example: Simplifying a Rational Expression

Let's consider the rational expression:

5k6⋅32k3\frac{5k}{6} \cdot \frac{3}{2k^3}

Our mission is to simplify this expression to its most reduced form. We'll embark on this journey step-by-step, carefully dissecting each stage of the process. By the end of this example, you'll have a clear understanding of the techniques involved and the logic behind each manipulation. So, let's roll up our sleeves and dive into the world of simplification!

Step 1: Multiplying Rational Expressions

When multiplying rational expressions, we follow a simple rule: multiply the numerators together and multiply the denominators together. This process transforms two fractions into a single fraction, setting the stage for simplification. In our example, we have:

5k6⋅32k3=5k⋅36⋅2k3=15k12k3\frac{5k}{6} \cdot \frac{3}{2k^3} = \frac{5k \cdot 3}{6 \cdot 2k^3} = \frac{15k}{12k^3}

Now, we have a single fraction that we can further simplify.

Step 2: Identifying Common Factors

The heart of simplifying rational expressions lies in identifying and canceling common factors. These are factors that appear in both the numerator and the denominator. In our expression, 15k12k3\frac{15k}{12k^3}, we can see that both the numerator and the denominator share common factors. Let's break down the numbers and variables to make these factors more apparent.

  • The numbers 15 and 12 share a common factor of 3.
  • The variable k appears in both the numerator and the denominator.

Step 3: Canceling Common Factors

Now comes the exciting part: canceling the common factors. This is where we divide both the numerator and the denominator by the common factors, effectively reducing the expression to its simplest form. Let's tackle the numerical factor first. We divide both 15 and 12 by 3:

15k12k3=15÷3otimesk12÷3otimesk3=5k4k3\frac{15k}{12k^3} = \frac{15 \div 3 otimes k}{12 \div 3 otimes k^3} = \frac{5k}{4k^3}

Next, we address the variable k. We have k in the numerator and k³ in the denominator. Using the rules of exponents, we can cancel out one k from both:

5k4k3=5otimesk14otimesk3=54k3−1=54k2\frac{5k}{4k^3} = \frac{5 otimes k^1}{4 otimes k^3} = \frac{5}{4k^{3-1}} = \frac{5}{4k^2}

And there you have it! We've successfully simplified the rational expression.

Step 4: The Simplified Expression

After canceling all common factors, we arrive at the simplified expression:

54k2\frac{5}{4k^2}

This is the most reduced form of the original expression. It's cleaner, more concise, and easier to work with. We've successfully navigated the simplification process, transforming a seemingly complex expression into a simple, elegant form.

Common Mistakes to Avoid

Simplifying rational expressions can be tricky, and it's easy to fall into common pitfalls. Here are some mistakes to watch out for:

  1. Canceling Terms Instead of Factors: Remember, you can only cancel factors, not terms. A factor is something that is multiplied, while a term is something that is added or subtracted. For example, in the expression x+22\frac{x + 2}{2}, you cannot cancel the 2s because they are part of a term (x + 2), not a factor.
  2. Forgetting to Factor First: Before you start canceling, make sure you've factored the numerator and denominator completely. This will help you identify all the common factors.
  3. Incorrectly Applying Exponent Rules: Pay close attention to the rules of exponents when canceling variables. Remember that xmxn=xm−n\frac{x^m}{x^n} = x^{m-n}.
  4. Missing Negative Signs: Be careful with negative signs. A negative sign can easily be overlooked, leading to an incorrect answer.
  5. Assuming Everything Cancels: Don't assume that everything will cancel out perfectly. Sometimes, the simplified expression will still have variables or constants in both the numerator and denominator.

By being aware of these common mistakes, you can avoid them and simplify rational expressions with greater accuracy and confidence.

Practice Problems

To solidify your understanding, let's tackle a few practice problems. These problems will give you the opportunity to apply the techniques we've discussed and hone your simplification skills.

  1. Simplify: 10x25x\frac{10x^2}{5x}
  2. Simplify: 3y9y4\frac{3y}{9y^4}
  3. Simplify: 2(a+b)4(a+b)2\frac{2(a + b)}{4(a + b)^2}

Take your time, work through each problem step-by-step, and remember to look for common factors. The more you practice, the more comfortable and proficient you'll become at simplifying rational expressions.

Conclusion

Simplifying rational expressions is a fundamental skill in algebra. By mastering this skill, you'll be able to tackle more complex mathematical problems with ease. Remember the key steps: multiply the expressions, identify common factors, cancel those factors, and arrive at the simplified form. Avoid common mistakes by factoring completely, canceling factors (not terms), and paying close attention to exponent rules and negative signs. With practice, you'll become a pro at simplifying rational expressions!

Answer to the Initial Problem

The simplified form of the expression 5k6⋅32k3\frac{5k}{6} \cdot \frac{3}{2k^3} is indeed 54k2\frac{5}{4k^2}, which corresponds to option A.