Simplifying Algebraic Expressions A Step-by-Step Guide

In this article, we will delve into the process of simplifying the given mathematical expression: $\frac{4 x^2+36}{4 x} \cdot \frac{1}{5 x}$. This type of problem often appears in algebra and requires a solid understanding of factoring, simplification, and algebraic manipulation. Our goal is to break down the expression step-by-step, ensuring clarity and comprehension for anyone looking to master these essential skills. We will explore each component of the expression, identify common factors, and perform the necessary simplifications to arrive at the most reduced form. By the end of this guide, you will not only be able to solve this specific problem but also gain a broader understanding of how to tackle similar algebraic challenges. Let's embark on this mathematical journey together and unlock the simplicity hidden within complex expressions.

Understanding the Expression

Before we dive into the simplification process, it's crucial to understand the structure of the expression $\frac{4 x^2+36}{4 x} \cdot \frac{1}{5 x}$. This expression involves the multiplication of two fractions. The first fraction, $rac{4 x^2+36}{4 x}$, has a numerator of $4x^2 + 36$ and a denominator of $4x$. The second fraction, $\frac{1}{5 x}$, is simpler with a numerator of 1 and a denominator of $5x$. To simplify this expression effectively, we'll need to focus on factoring, canceling common factors, and then combining the remaining terms. The numerator of the first fraction, $4x^2 + 36$, is a key area where factoring can significantly simplify the expression. We should also pay close attention to the denominators, as any common factors between the numerators and denominators can be canceled out. Remember, the ultimate goal of simplifying is to reduce the expression to its most basic form, making it easier to understand and work with in further calculations or applications. This initial understanding sets the stage for a methodical approach to simplification, ensuring we don't miss any crucial steps.

Factoring the Numerator

The first critical step in simplifying the expression $\frac{4 x^2+36}{4 x} \cdot \frac{1}{5 x}$ is to factor the numerator of the first fraction, which is $4x^2 + 36$. Factoring is the process of breaking down an expression into its constituent multiplicative parts. In this case, we observe that both terms in the numerator, $4x^2$ and $36$, share a common factor of 4. Factoring out the 4, we rewrite the numerator as $4(x^2 + 9)$. This step is crucial because it allows us to potentially cancel out common factors with the denominator later on. The expression $x^2 + 9$ is a sum of squares, and in the realm of real numbers, it cannot be factored further. However, recognizing and extracting the common factor of 4 significantly simplifies the overall expression. By factoring the numerator, we transform the original expression into $\frac{4(x^2+9)}{4 x} \cdot \frac{1}{5 x}$, which sets the stage for the next phase of simplification: canceling common factors. This methodical approach ensures we handle each part of the expression with care, leading to an accurate and simplified result.

Canceling Common Factors

Now that we have factored the numerator, the expression looks like this: $\frac{4(x^2+9)}{4 x} \cdot \frac{1}{5 x}$. The next crucial step is to cancel any common factors between the numerators and the denominators. By doing so, we can reduce the complexity of the expression significantly. Looking at the first fraction, we can see that the number 4 appears in both the numerator and the denominator. This means we can cancel out the 4, simplifying the fraction. After canceling the 4, the expression becomes $\frac{x^2+9}{x} \cdot \frac{1}{5 x}$. It’s important to note that we can only cancel factors that are common to both the numerator and the denominator; we cannot cancel terms that are added or subtracted. In this case, $x^2 + 9$ is treated as a single term, and since it doesn't have any common factors with the denominators, it remains as is. This cancellation step is a fundamental technique in simplifying algebraic expressions, and mastering it is essential for tackling more complex problems. With the common factor of 4 removed, we are one step closer to the simplest form of our expression.

Multiplying the Fractions

With the common factors canceled, we now have the expression $\frac{x^2+9}{x} \cdot \frac{1}{5 x}$. The next step in simplifying is to multiply the fractions. To do this, we multiply the numerators together and the denominators together. Multiplying the numerators, we have $(x^2 + 9) \cdot 1$, which simply equals $x^2 + 9$. Multiplying the denominators, we have $x \cdot 5x$, which equals $5x^2$. Therefore, the result of multiplying the fractions is $\frac{x^2+9}{5 x^2}$. This step combines the two fractions into a single fraction, making it easier to see if further simplification is possible. The process of multiplying fractions is straightforward, but it's crucial to ensure that you multiply the correct terms and keep track of the resulting expression. Now that we have a single fraction, we can assess whether any further simplification can be done. In this case, the numerator $x^2 + 9$ cannot be factored further over the real numbers, and there are no common factors between the numerator and the denominator. Thus, we have reached the simplest form of the expression.

Final Simplified Expression

After performing all the necessary steps – factoring, canceling common factors, and multiplying the fractions – we have arrived at the final simplified expression. From our previous steps, we obtained the fraction $\frac{x^2+9}{5 x^2}$. We've already established that the numerator, $x^2 + 9$, cannot be factored further over the real numbers, and there are no common factors between the numerator and the denominator. This means that the fraction is in its simplest form. Therefore, the simplified form of the original expression $\frac{4 x^2+36}{4 x} \cdot \frac{1}{5 x}$ is $\frac{x^2+9}{5 x^2}$. This final result is concise and easy to understand, which is the ultimate goal of simplification. It's important to double-check your work to ensure that you haven't missed any potential simplifications, but in this case, we can confidently say that we have reached the simplest form. This step-by-step process illustrates how complex algebraic expressions can be broken down and simplified with careful attention to detail and a solid understanding of algebraic principles.

Conclusion

In conclusion, we have successfully simplified the expression $\frac{4 x^2+36}{4 x} \cdot \frac{1}{5 x}$ by following a systematic approach. We began by understanding the structure of the expression and identifying the key areas for simplification. The next step involved factoring the numerator of the first fraction, which allowed us to reveal a common factor. We then proceeded to cancel common factors between the numerators and denominators, significantly reducing the complexity of the expression. Following this, we multiplied the fractions to combine them into a single fraction. Finally, we arrived at the simplified expression $\frac{x^2+9}{5 x^2}$, which represents the most reduced form of the original expression. This exercise demonstrates the importance of mastering algebraic techniques such as factoring, canceling, and multiplying fractions. By breaking down complex problems into smaller, manageable steps, we can simplify even the most challenging expressions. This methodical approach not only helps in arriving at the correct answer but also enhances our understanding of the underlying mathematical principles. Simplifying algebraic expressions is a fundamental skill in mathematics, and the ability to do so is crucial for success in higher-level courses and real-world applications.

Answer

The correct answer is B) $\frac{x^2+9}{5 x^2}$.