Simplifying Algebraic Expressions Perform Indicated Operations

Jul 11, 2025 by Admin 63 views

Perform the Indicated Operations and Reduce to Lowest Terms: A Comprehensive Guide

\frac{5 x^2-45}{x^6} \div \frac{x-3}{x^4}

In the realm of mathematics, simplifying expressions is a fundamental skill. This article provides a step-by-step guide to performing the indicated operations and reducing the expression $\frac{5 x^2-45}{x6} \div \frac{x-3}{x^4}$ to its lowest terms. We will assume that no denominator has a value of zero throughout the process. Mastery of this concept is crucial for success in algebra and calculus. To effectively perform operations and reduce expressions, it's vital to have a solid grasp of factoring, simplifying fractions, and handling exponents. These skills are the building blocks for more advanced mathematical concepts. This article breaks down each step, ensuring a clear understanding of the process involved in simplifying complex algebraic fractions. Let’s dive into the solution.

Step 1: Convert Division to Multiplication

The first step in simplifying the given expression is to convert the division operation into multiplication. Remember that dividing by a fraction is the same as multiplying by its reciprocal. Therefore, we can rewrite the expression as follows:

\frac{5 x^2-45}{x^6} \div \frac{x-3}{x^4} = \frac{5 x^2-45}{x^6} \times \frac{x^4}{x-3}

This transformation is crucial because it allows us to combine the fractions into a single expression, making the simplification process more manageable. By changing the division to multiplication, we set the stage for factoring and canceling common terms, which are essential steps in reducing the expression to its simplest form. Understanding this initial step is key to solving similar problems in algebra. The reciprocal of a fraction is obtained by swapping the numerator and the denominator. This basic principle is essential for manipulating and simplifying algebraic expressions involving division.

Step 2: Factor the Numerator

Next, we need to factor the numerator of the first fraction, which is $5x^2 - 45$. We can start by factoring out the greatest common factor (GCF), which is 5:

5 x^2 - 45 = 5(x^2 - 9)

Now, notice that $x^2 - 9$ is a difference of squares, which can be further factored as $(x - 3)(x + 3)$. Thus, the factored form of the numerator is:

5(x^2 - 9) = 5(x - 3)(x + 3)

Factoring is a critical skill in simplifying algebraic expressions. It allows us to identify common factors that can be canceled out, leading to a simplified form. In this case, recognizing the difference of squares pattern is key to fully factoring the numerator. Mastering factoring techniques is essential for success in algebra and beyond. The ability to factor expressions quickly and accurately can significantly reduce the complexity of many mathematical problems. Factoring not only simplifies expressions but also helps in solving equations and understanding the behavior of functions.

Step 3: Rewrite the Expression with Factored Terms

Now that we have factored the numerator, we can rewrite the entire expression with the factored terms:

\frac{5(x - 3)(x + 3)}{x^6} \times \frac{x^4}{x - 3}

This step is crucial because it clearly shows all the factors in the expression, making it easier to identify common terms that can be canceled out. Rewriting the expression in this form sets the stage for the next step, which involves canceling common factors. By expressing the numerator and denominator in their factored forms, we gain a clearer understanding of the structure of the expression and can proceed with simplification more effectively. This process highlights the importance of factoring in simplifying complex algebraic expressions.

Step 4: Cancel Common Factors

Now, we can cancel out the common factors between the numerator and the denominator. We have a factor of $(x - 3)$ in both the numerator and the denominator, which can be canceled:

\frac{5(x - 3)(x + 3)}{x^6} \times \frac{x^4}{x - 3} = \frac{5(x + 3)}{x^6} \times x^4

Additionally, we can simplify the powers of $x$. We have $x^4$ in the numerator and $x^6$ in the denominator. When dividing powers with the same base, we subtract the exponents:

\frac{x^4}{x^6} = \frac{1}{x^{6-4}} = \frac{1}{x^2}

So, the expression becomes:

\frac{5(x + 3)}{x^6} \times x^4 = \frac{5(x + 3)}{x^2}

Canceling common factors is a fundamental step in simplifying algebraic expressions. It involves identifying factors that appear in both the numerator and the denominator and removing them. This process reduces the expression to its simplest form. By canceling common factors, we eliminate unnecessary complexity and make the expression easier to work with. This step demonstrates the power of factoring in simplifying mathematical expressions. The ability to identify and cancel common factors is a crucial skill for solving algebraic problems.

Step 5: Write the Simplified Expression

Finally, we have simplified the expression to its lowest terms:

\frac{5(x + 3)}{x^2}

This is the simplified form of the original expression. We have performed the indicated operations and reduced the expression to its lowest terms by factoring, canceling common factors, and simplifying exponents. This final step represents the culmination of the simplification process. The simplified expression is easier to understand and work with than the original expression. By following these steps, we have successfully reduced a complex algebraic fraction to its simplest form. The ability to simplify expressions is a crucial skill in mathematics, allowing us to solve problems more efficiently and accurately.

Detailed Step-by-Step Solution

Let's recap the steps we took to solve this problem:

Convert Division to Multiplication: We started by converting the division operation to multiplication by taking the reciprocal of the second fraction.
$\frac{5 x^2-45}{x^6} \div \frac{x-3}{x^4} = \frac{5 x^2-45}{x^6} \times \frac{x^4}{x-3}$
Factor the Numerator: We factored the numerator of the first fraction.
$5 x^2 - 45 = 5(x^2 - 9) = 5(x - 3)(x + 3)$
Rewrite the Expression with Factored Terms: We rewrote the expression with the factored terms.
$\frac{5(x - 3)(x + 3)}{x^6} \times \frac{x^4}{x - 3}$
Cancel Common Factors: We canceled out the common factors between the numerator and the denominator.
$\frac{5(x - 3)(x + 3)}{x^6} \times \frac{x^4}{x - 3} = \frac{5(x + 3)}{x^2}$
Write the Simplified Expression: We wrote the simplified expression.
$\frac{5(x + 3)}{x^2}$

Each of these steps is crucial in simplifying the expression. By following this methodical approach, you can tackle similar problems with confidence. This detailed step-by-step solution provides a clear roadmap for simplifying algebraic expressions. Understanding each step is essential for mastering the process and applying it to other problems. The ability to break down a complex problem into smaller, manageable steps is a key skill in mathematics.

Common Mistakes to Avoid

When simplifying algebraic expressions, it's easy to make mistakes. Here are some common pitfalls to watch out for:

Forgetting to Factor Completely: Always ensure that you have factored the expression completely before canceling terms. Missing a factor can lead to an incorrect simplification.
Incorrectly Canceling Terms: You can only cancel factors, not terms. For example, you cannot cancel $x$ in the expression $\frac{x + 3}{x}$. This is a common mistake that can lead to incorrect answers. Understanding the difference between factors and terms is crucial for correct simplification.
Errors with Exponents: When simplifying exponents, remember the rules of exponents. For example, when dividing powers with the same base, you subtract the exponents.
Not Converting Division to Multiplication: Failing to convert division to multiplication by the reciprocal can lead to errors in the simplification process. This initial step is crucial for correctly simplifying expressions involving division.
Arithmetic Errors: Simple arithmetic errors can derail the entire simplification process. Double-check your calculations to avoid these mistakes.

Avoiding these common mistakes will help you simplify algebraic expressions accurately and efficiently. Attention to detail and a thorough understanding of the rules of algebra are essential for success. By being aware of these potential pitfalls, you can minimize errors and improve your problem-solving skills. Practicing and reviewing these common mistakes can help you build confidence and accuracy in your mathematical work.

Practice Problems

To solidify your understanding, here are some practice problems:

$\frac{x^2 - 4}{x^2 - 4x + 4} \div \frac{x + 2}{x - 2}$
$\frac{3y^2 - 12}{y^2} \times \frac{y}{y - 2}$
$\frac{2z^2 + 4z}{z^3} \div \frac{z + 2}{z^2}$

Work through these problems, applying the steps we've discussed. Check your answers to ensure you've correctly simplified the expressions. Practice is essential for mastering any mathematical skill. By working through these problems, you can reinforce your understanding of the concepts and techniques involved in simplifying algebraic expressions. The more you practice, the more confident and proficient you will become. These practice problems provide an opportunity to apply the knowledge you've gained and develop your problem-solving skills.

Conclusion

Simplifying algebraic expressions involves several key steps, including converting division to multiplication, factoring, canceling common factors, and simplifying exponents. By following these steps carefully and avoiding common mistakes, you can successfully reduce expressions to their lowest terms. Mastering these skills is crucial for success in algebra and beyond. This comprehensive guide has provided you with the knowledge and tools necessary to tackle these types of problems with confidence. Remember to practice regularly and review the concepts as needed. The ability to simplify algebraic expressions is a fundamental skill that will serve you well in your mathematical journey. Consistent practice and a solid understanding of the underlying principles are the keys to success.