Simplifying The Expression 3/(xy^2) + 5/(x^2y) A Step-by-Step Guide

In the realm of algebra, simplifying expressions is a fundamental skill. Among these expressions, fractions involving variables, also known as algebraic fractions, often present a challenge. This article delves into the simplification of a specific algebraic fraction: $\frac{3}{xy^2} + \frac{5}{x^2y}$. We will break down the process step by step, ensuring clarity and understanding for learners of all levels.

Understanding the Basics of Algebraic Fractions

Before diving into the specifics of our example, let's establish a solid foundation in algebraic fractions. Algebraic fractions, much like their numerical counterparts, represent a part of a whole, but in this case, the whole is expressed using variables and constants. The key to simplifying these fractions lies in finding a common denominator, a principle that mirrors the simplification of numerical fractions.

The core concept revolves around the idea that you can only add or subtract fractions if they share the same denominator. This shared denominator allows us to combine the numerators while keeping the denominator consistent. When dealing with algebraic fractions, the common denominator will often involve finding the least common multiple (LCM) of the denominators' variable and constant components.

To effectively simplify algebraic fractions, you must understand key concepts, including factors, multiples, and the distributive property. Factoring helps break down expressions into their simplest components, while understanding multiples allows us to identify the LCM. The distributive property comes into play when we need to multiply terms across parentheses, a common occurrence in simplification processes. It is useful to remember that you must multiply the numerator by the same value that was used to multiply the denominator to maintain the integrity of the fraction. For example, if you multiply the denominator by x, you must also multiply the numerator by x.

Consider the analogy of adding numerical fractions like $\frac{1}{2} + \frac{1}{3}$. We can't directly add these because they have different denominators. We find the least common multiple (LCM) of 2 and 3, which is 6. Then, we convert both fractions to have a denominator of 6: $\frac{1}{2}$ becomes $\frac{3}{6}$ and $\frac{1}{3}$ becomes $\frac{2}{6}$. Now we can add them: $\frac{3}{6} + \frac{2}{6} = \frac{5}{6}$. The same principle applies to algebraic fractions, but instead of dealing with numbers, we are working with variables and their exponents.

Step-by-Step Simplification of $\frac{3}{xy^2} + \frac{5}{x^2y}$

Now, let's tackle the problem at hand: simplifying $\frac{3}{xy^2} + \frac{5}{x^2y}$. We will follow a structured approach, breaking the process into manageable steps.

1. Identify the Denominators

The first step is to clearly identify the denominators of the fractions we are working with. In this case, we have two fractions: $\frac{3}{xy^2}$ and $\frac{5}{x^2y}$. The denominators are $xy^2$ and $x^2y$, respectively. It is important to visually separate and understand each denominator as a unique expression.

2. Find the Least Common Denominator (LCD)

Next, we need to determine the least common denominator (LCD). The LCD is the smallest expression that is a multiple of both denominators. To find the LCD, we consider the variables and their highest powers present in each denominator.

Looking at our denominators, $xy^2$ and $x^2y$, we see that the variables involved are x and y. The highest power of x is $x^2$ (from the second denominator), and the highest power of y is $y^2$ (from the first denominator). Therefore, the LCD is the product of these highest powers: $x^2y^2$.

Understanding how to find the LCD is crucial for simplifying algebraic fractions. It ensures that we are working with the smallest possible common denominator, which simplifies the subsequent steps. This process involves examining the variables and exponents in each denominator and combining them to create the LCD. If there are numerical coefficients in the denominator, you'll also need to find the least common multiple of those coefficients.

3. Convert Fractions to Equivalent Fractions with the LCD

Now, we need to convert each fraction into an equivalent fraction that has the LCD ($x^2y^2$) as its denominator. To do this, we multiply the numerator and denominator of each fraction by the appropriate factors.

For the first fraction, $\frac{3}{xy^2}$, we need to multiply the denominator by x to get $x^2y^2$. To maintain the fraction's value, we must also multiply the numerator by x. This gives us:

$\frac{3}{xy^2} * \frac{x}{x} = \frac{3x}{x^2y^2}$

For the second fraction, $\frac{5}{x^2y}$, we need to multiply the denominator by y to get $x^2y^2$. Again, we multiply the numerator by the same factor:

$\frac{5}{x^2y} * \frac{y}{y} = \frac{5y}{x^2y^2}$

At this point, both fractions have the same denominator, the LCD of $x^2y^2$. This is a critical step, as it allows us to combine the numerators in the next step.

4. Add the Numerators

With the fractions now sharing a common denominator, we can add the numerators together. This is a straightforward process, as we simply add the expressions in the numerators while keeping the denominator the same.

We have $\frac{3x}{x^2y^2}$ and $\frac{5y}{x^2y^2}$. Adding the numerators, we get:

$\frac{3x + 5y}{x^2y^2}$

The resulting fraction, $\frac{3x + 5y}{x^2y^2}$, represents the sum of the two original fractions. At this stage, we have combined the two fractions into a single fraction with the common denominator.

5. Simplify the Result (if possible)

The final step is to check if the resulting fraction can be further simplified. Simplification involves looking for common factors in the numerator and denominator that can be canceled out. In this case, our fraction is:

$\frac{3x + 5y}{x^2y^2}$

Looking at the numerator ($3x + 5y$) and the denominator ($x^2y^2$), we see that there are no common factors that can be canceled. The terms in the numerator, $3x$ and $5y$, are distinct and do not share any factors with the denominator. The denominator consists of powers of x and y, but these do not directly factor out of the numerator's expression.

Therefore, the fraction $\frac{3x + 5y}{x^2y^2}$ is already in its simplest form. There are no further steps we can take to reduce it.

The Solution

By following these steps, we have successfully simplified the expression $\frac{3}{xy^2} + \frac{5}{x^2y}$. The simplified form is:

$\frac{3x + 5y}{x^2y^2}$

This matches one of the options provided. The correct response is:

$\frac{3x+5y}{x^2 y^2}$

Common Mistakes to Avoid

When simplifying algebraic fractions, several common mistakes can arise. Being aware of these pitfalls can help you avoid them.

• Incorrectly Identifying the LCD: A frequent error is miscalculating the least common denominator. Remember to consider the highest powers of each variable present in the denominators. Forgetting to do so can lead to an incorrect common denominator, which will throw off the entire simplification process.
• Adding Numerators Without a Common Denominator: This is a fundamental mistake in fraction addition. You can only add fractions once they have the same denominator. Adding numerators directly without a common denominator will result in an incorrect answer.
• Incorrectly Multiplying to Obtain Equivalent Fractions: When converting fractions to equivalent fractions with the LCD, ensure you multiply both the numerator and denominator by the same factor. Multiplying only one part changes the fraction's value and leads to errors.
• Failure to Simplify the Final Result: Always check if the final fraction can be simplified further by canceling common factors. Leaving a fraction in a non-simplified form means the simplification process is incomplete.
• Distributive Property Errors: When the numerator involves multiple terms, correctly apply the distributive property if there are multiplications involved. A mistake in distribution can lead to an incorrect numerator and, consequently, an incorrect simplified fraction.

Conclusion

Simplifying algebraic fractions is a crucial skill in algebra. By understanding the fundamental principles, like finding the least common denominator and converting fractions to equivalent forms, you can confidently tackle these problems. This step-by-step guide, using the example of $\frac{3}{xy^2} + \frac{5}{x^2y}$, provides a clear roadmap for simplifying such expressions. Remember to practice consistently, pay attention to detail, and avoid common mistakes. With dedication and a systematic approach, simplifying algebraic fractions can become a manageable and even enjoyable part of your mathematical journey.