Simplify: 4/(9wy³) + 5/(12w²y) - Step-by-Step Solution

Let's tackle the simplification of the expression $\frac{4}{9 w y^3}+\frac{5}{12 w^2 y}$. This involves finding a common denominator and combining the fractions. Simplifying algebraic fractions is a fundamental skill in mathematics, and understanding how to manipulate these expressions is crucial for more advanced topics. This article will guide you through each step of the process, ensuring you grasp the underlying principles and can confidently apply them to similar problems.

Finding the Least Common Denominator (LCD)

To add these two fractions, we first need to find the least common denominator (LCD). The LCD is the smallest expression that both denominators ($9wy^3$ and $12w^2y$) divide into evenly. Let’s break down each denominator into its prime factors and variables:

• $9wy^3 = 3^2 * w * y^3$
• $12w^2y = 2^2 * 3 * w^2 * y$

Now, to find the LCD, we take the highest power of each prime factor and variable present in either denominator:

• The highest power of 2 is $2^2 = 4$.
• The highest power of 3 is $3^2 = 9$.
• The highest power of w is $w^2$.
• The highest power of y is $y^3$.

Multiplying these together, we get the LCD: $4 * 9 * w^2 * y^3 = 36w^2y^3$. Understanding the process of finding the LCD is essential. The LCD ensures that we can combine the fractions correctly, as it provides a common base that both original denominators can be transformed into. Without a common denominator, adding fractions becomes meaningless, similar to adding apples and oranges. The LCD method guarantees that we are working with comparable units, allowing for accurate summation.

Adjusting the Fractions

Now that we have the LCD, we need to adjust each fraction so that it has this denominator. To do this, we multiply each fraction by a form of 1 that will change the denominator to $36w^2y^3$.

For the first fraction, $\frac{4}{9wy^3}$, we need to multiply the denominator by $4w$ to get $36w^2y^3$. Therefore, we multiply both the numerator and the denominator by $4w$:

$\frac{4}{9wy^3} * \frac{4w}{4w} = \frac{16w}{36w^2y^3}$

For the second fraction, $\frac{5}{12w^2y}$, we need to multiply the denominator by $3y^2$ to get $36w^2y^3$. So, we multiply both the numerator and the denominator by $3y^2$:

$\frac{5}{12w^2y} * \frac{3y^2}{3y^2} = \frac{15y^2}{36w^2y^3}$

The key to this step is ensuring that we multiply both the numerator and denominator by the same expression. This maintains the value of the fraction while allowing us to achieve a common denominator. This is a crucial step because it sets the stage for combining the fractions in the next step. By manipulating the fractions in this way, we are essentially rewriting them in an equivalent form that makes addition straightforward. The ability to adjust fractions to a common denominator is a cornerstone of algebraic manipulation.

Combining the Fractions

Now that both fractions have the same denominator, we can add them together. We simply add the numerators and keep the denominator the same:

$\frac{16w}{36w^2y^3} + \frac{15y^2}{36w^2y^3} = \frac{16w + 15y^2}{36w^2y^3}$

This step is relatively straightforward once the fractions have a common denominator. We are simply combining like terms in the numerator. It's important to remember that we can only add fractions when they have the same denominator; otherwise, we would be adding different units. In this case, we are adding the numerators, which represent the number of parts we have, while the denominator represents the size of each part. Adding fractions with a common denominator is a fundamental arithmetic operation that extends naturally to algebraic expressions.

Checking for Further Simplification

Finally, we should check to see if the resulting fraction can be simplified further. In this case, the numerator ($16w + 15y^2$) and the denominator ($36w^2y^3$) do not share any common factors other than 1. Therefore, the fraction is already in its simplest form.

Thus, the simplified expression is:

$\frac{16w + 15y^2}{36w^2y^3}$

Always remember to check for further simplification. This ensures that you have expressed the fraction in its most reduced form. Identifying and canceling out common factors between the numerator and denominator is an essential step in simplifying algebraic expressions. While in this case, no further simplification is possible, it's a good practice to always verify to ensure the final answer is in its simplest form. This skill is particularly important when working with more complex expressions where simplification may not be immediately obvious.

Conclusion

In summary, to simplify the expression $\frac{4}{9 w y^3}+\frac{5}{12 w^2 y}$, we followed these steps:

1. Found the least common denominator (LCD), which was $36w^2y^3$.
2. Adjusted each fraction to have the LCD as its denominator.
3. Combined the fractions by adding the numerators.
4. Checked for further simplification.

The final simplified expression is $\frac{16w + 15y^2}{36w^2y^3}$.

Understanding and mastering these steps will enable you to simplify a wide range of algebraic fractions. Remember to always look for the LCD, adjust the fractions accordingly, combine them, and check for any possible simplifications at the end. Practice is key to becoming proficient in simplifying algebraic expressions. The more you practice, the more comfortable and confident you will become in applying these techniques.

For further learning and practice on simplifying algebraic expressions, you can visit Khan Academy's Algebra section for comprehensive lessons and exercises.