Simplifying (ab)/(3a-6b) * (4a-8b)/(a^2b) A Step-by-Step Guide

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In the realm of mathematics, simplifying rational expressions is a fundamental skill, crucial for tackling more advanced algebraic problems. Rational expressions, essentially fractions with polynomials in the numerator and denominator, often appear complex at first glance. However, by applying a systematic approach involving factoring, canceling common factors, and understanding restrictions, we can effectively simplify these expressions. This article will delve into the process of simplifying the rational expression (ab)/(3a-6b) * (4a-8b)/(a^2b), providing a step-by-step guide suitable for learners of all levels.

Understanding the Basics of Rational Expressions

Before we dive into the simplification process, it's essential to grasp the basic concepts of rational expressions. A rational expression is any expression that can be written in the form P/Q, where P and Q are polynomials, and Q is not equal to zero. The restriction that Q cannot be zero is critical because division by zero is undefined in mathematics. Understanding this restriction is key to correctly simplifying and interpreting rational expressions. The ability to identify these restrictions is crucial for defining the domain of the rational expression, which is the set of all possible values that the variable(s) can take without making the denominator zero. Simplifying rational expressions often involves identifying common factors in the numerator and the denominator and canceling them out. This process is analogous to simplifying numerical fractions, where we divide both the numerator and denominator by their greatest common divisor. However, with rational expressions, we are dealing with polynomials, so the factoring process is more involved.

To successfully simplify rational expressions, a solid understanding of factoring techniques is essential. Factoring is the process of breaking down a polynomial into its constituent factors, which are simpler polynomials that, when multiplied together, give the original polynomial. Common factoring techniques include factoring out the greatest common factor (GCF), factoring quadratic expressions, and using special factoring patterns such as the difference of squares or the sum/difference of cubes. Mastering these factoring techniques is crucial for simplifying rational expressions efficiently. In the context of simplifying (ab)/(3a-6b) * (4a-8b)/(a^2b), we will employ factoring techniques to identify common factors in both the numerator and the denominator. This will allow us to cancel out these factors and arrive at the simplified form of the expression. The process of simplification not only makes the expression easier to work with but also reveals the underlying structure and relationships between the variables involved. It's a skill that's applicable in various areas of mathematics, including calculus, algebra, and trigonometry.

Step-by-Step Simplification of (ab)/(3a-6b) * (4a-8b)/(a^2b)

Now, let's proceed with the step-by-step simplification of the given rational expression:

(ab)/(3a-6b) * (4a-8b)/(a^2b)

Step 1: Factoring

The first step in simplifying this rational expression is to factor both the numerators and denominators of the fractions. This will allow us to identify any common factors that can be canceled out. Factoring is a fundamental technique in algebra, and proficiency in factoring is crucial for simplifying rational expressions.

  • Numerator of the first fraction: The numerator is simply ab, which is already in its simplest factored form.
  • Denominator of the first fraction: The denominator is 3a - 6b. We can factor out the greatest common factor (GCF), which is 3. This gives us 3(a - 2b).
  • Numerator of the second fraction: The numerator is 4a - 8b. Again, we can factor out the GCF, which is 4. This gives us 4(a - 2b).
  • Denominator of the second fraction: The denominator is a^2b. This is already in a factored form, but we can rewrite it as a * a * b to make it clearer for the next step.

After factoring, our expression looks like this:

(ab) / [3(a - 2b)] * [4(a - 2b)] / (a^2b)

Step 2: Multiplying the Fractions

Next, we multiply the two fractions together. To do this, we multiply the numerators and the denominators separately:

[ab * 4(a - 2b)] / [3(a - 2b) * a^2b]

This simplifies to:

[4ab(a - 2b)] / [3a^2b(a - 2b)]

Step 3: Canceling Common Factors

Now, we identify and cancel out any common factors that appear in both the numerator and the denominator. This is the key step in simplifying rational expressions. Look for factors that are identical in both the numerator and the denominator.

  • We have ab in the numerator and a^2b in the denominator. We can cancel out ab from both, leaving us with a in the denominator.
  • We also have (a - 2b) in both the numerator and the denominator. We can cancel this out completely.

After canceling the common factors, our expression becomes:

4 / (3a)

Step 4: Final Simplified Form

Therefore, the simplified form of the rational expression (ab)/(3a-6b) * (4a-8b)/(a^2b) is:

4 / (3a)

This is the final simplified form of the given expression. We have successfully factored, multiplied, and canceled common factors to arrive at this result. The process demonstrates the importance of factoring and identifying common factors in simplifying rational expressions.

Restrictions on the Variables

As mentioned earlier, it is crucial to consider the restrictions on the variables when working with rational expressions. Restrictions arise from the fact that division by zero is undefined. Therefore, any values of the variables that make the denominator equal to zero must be excluded from the domain of the expression. Identifying these restrictions is an important part of simplifying rational expressions.

In our original expression, (ab)/(3a-6b) * (4a-8b)/(a^2b), we had two denominators: 3a - 6b and a^2b. Let's analyze each of these to find the restrictions.

Restriction from 3a - 6b

We need to find the values of a and b that make 3a - 6b = 0. We can set up the equation:

3a - 6b = 0

Divide both sides by 3:

a - 2b = 0

Add 2b to both sides:

a = 2b

This means that if a = 2b, the denominator 3a - 6b will be zero, and the expression will be undefined. Therefore, a cannot be equal to 2b.

Restriction from a^2b

We also need to find the values of a and b that make a^2b = 0. This occurs when either a = 0 or b = 0 (or both). If a = 0 or b = 0, the denominator a^2b will be zero, and the expression will be undefined. Therefore, a cannot be 0 and b cannot be 0.

Summary of Restrictions

In summary, the restrictions on the variables for the expression (ab)/(3a-6b) * (4a-8b)/(a^2b) are:

  • a ≠ 2b
  • a ≠ 0
  • b ≠ 0

These restrictions are important to keep in mind when working with the simplified form of the expression, 4 / (3a). Even though the denominator in the simplified form is only 3a, we must still adhere to the original restrictions to ensure the equivalence of the expressions. The simplified expression is only valid for values of a and b that satisfy these restrictions. Understanding and stating these restrictions is a crucial aspect of simplifying rational expressions completely and accurately.

Common Mistakes and How to Avoid Them

When simplifying rational expressions, it's easy to make mistakes if you're not careful. Here are some common errors and tips on how to avoid them:

  1. Incorrect Factoring: Factoring is a crucial first step, and any errors here will propagate through the rest of the problem. Always double-check your factoring by multiplying the factors back together to ensure you get the original expression. Look for the greatest common factor (GCF) first, and be mindful of special factoring patterns like the difference of squares or perfect square trinomials.

  2. Canceling Terms Instead of Factors: One of the most common mistakes is canceling terms that are not factors. Remember, you can only cancel factors that are multiplied together, not terms that are added or subtracted. For example, in the expression (a + b) / a, you cannot cancel the a in the numerator and denominator because it's a term, not a factor. Only cancel factors, which are expressions that are multiplied together.

  3. Forgetting Restrictions: As we've discussed, identifying and stating the restrictions on the variables is essential. Forgetting to do this can lead to incorrect solutions and a misunderstanding of the domain of the expression. Always identify the restrictions by setting the original denominators equal to zero and solving for the variables.

  4. Not Simplifying Completely: Sometimes, you might cancel some common factors but miss others. Always make sure you've canceled all common factors before arriving at the final simplified form. Double-check your work to ensure there are no more common factors in the numerator and denominator.

  5. Making Sign Errors: Sign errors are easy to make, especially when factoring or distributing negative signs. Pay close attention to signs throughout the problem, and double-check your work to catch any mistakes.

  6. Incorrectly Applying the Distributive Property: When multiplying fractions, make sure you distribute correctly if there are expressions in parentheses. For example, when multiplying 2(x + 3), make sure you distribute the 2 to both x and 3. Carefully apply the distributive property to avoid errors.

By being aware of these common mistakes and taking steps to avoid them, you can improve your accuracy and confidence in simplifying rational expressions. Practice is key, so work through plenty of examples and double-check your work to reinforce these concepts.

Conclusion

In conclusion, simplifying rational expressions is a crucial skill in algebra. By following a systematic approach that includes factoring, multiplying fractions, canceling common factors, and understanding restrictions, you can effectively simplify complex expressions. The example we worked through, (ab)/(3a-6b) * (4a-8b)/(a^2b), illustrates the process clearly. Remember to always factor first, multiply the numerators and denominators, cancel common factors, and state the restrictions on the variables. Avoiding common mistakes, such as canceling terms instead of factors or forgetting restrictions, will help you achieve accurate results. Mastering the art of simplifying rational expressions not only enhances your algebraic skills but also provides a solid foundation for more advanced mathematical concepts.

Simplifying rational expressions is more than just a mechanical process; it's a testament to the beauty and structure of mathematics. By understanding the underlying principles and practicing consistently, you can unlock the power of algebra and solve a wide range of problems. So, embrace the challenge, hone your skills, and enjoy the journey of mathematical exploration.