Add: $\frac{5}{6 C^5 D^2} + \frac{2}{18 C^5 D^3}$

Introduction

When adding fractions with different denominators, it can be challenging to determine the common denominator. However, with a clear understanding of the concept and a step-by-step approach, you can easily add fractions with different denominators. In this article, we will explore the process of adding fractions with different denominators using the given example: $\frac{5}{6 c^5 d^2} + \frac{2}{18 c^5 d^3}$.

Understanding the Concept of Common Denominator

Before we dive into the example, let's understand the concept of a common denominator. A common denominator is the least common multiple (LCM) of the denominators of two or more fractions. It is the smallest number that both denominators can divide into evenly. In the given example, the denominators are $6 c^5 d^2$ and $18 c^5 d^3$. To find the common denominator, we need to find the LCM of these two expressions.

Finding the Least Common Multiple (LCM)

To find the LCM of $6 c^5 d^2$ and $18 c^5 d^3$, we need to factorize both expressions. The factorization of $6 c^5 d^2$ is $2 \cdot 3 \cdot c^5 \cdot d^2$, and the factorization of $18 c^5 d^3$ is $2 \cdot 3^2 \cdot c^5 \cdot d^3$. The LCM of these two expressions is the product of the highest power of each factor that appears in either expression. Therefore, the LCM is $2 \cdot 3^2 \cdot c^5 \cdot d^3$.

Rewriting the Fractions with the Common Denominator

Now that we have found the common denominator, we can rewrite both fractions with the common denominator. To do this, we need to multiply the numerator and denominator of each fraction by the necessary factors to obtain the common denominator. For the first fraction, we need to multiply the numerator and denominator by $3 d^3$ to obtain the common denominator. For the second fraction, we need to multiply the numerator and denominator by $d^2$ to obtain the common denominator.

\frac{5}{6 c^5 d^2} = \frac{5 \cdot 3 d^3}{6 c^5 d^2 \cdot 3 d^3} = \frac{15 d^3}{18 c^5 d^5}
\frac{2}{18 c^5 d^3} = \frac{2 \cdot d^2}{18 c^5 d^3 \cdot d^2} = \frac{2 d^2}{18 c^5 d^5}

Adding the Fractions

Now that we have rewritten both fractions with the common denominator, we can add them. To do this, we simply add the numerators and keep the common denominator.

\frac{15 d^3}{18 c^5 d^5} + \frac{2 d^2}{18 c^5 d^5} = \frac{15 d^3 + 2 d^2}{18 c^5 d^5}

Simplifying the Result

The final step is to simplify the result. To do this, we can factor out the greatest common factor (GCF) of the numerator. In this case, the GCF is $d^2$. Therefore, we can factor out $d^2$ from the numerator.

\frac{15 d^3 + 2 d^2}{18 c^5 d^5} = \frac{d^2 (15 d + 2)}{18 c^5 d^5}

Conclusion

In this article, we have explored the process of adding fractions with different denominators using the given example: $\frac{5}{6 c^5 d^2} + \frac{2}{18 c^5 d^3}$. We have found the common denominator, rewritten both fractions with the common denominator, added the fractions, and simplified the result. With this step-by-step guide, you can easily add fractions with different denominators and simplify the result.

Common Mistakes to Avoid

When adding fractions with different denominators, there are several common mistakes to avoid. Here are a few:

• Not finding the common denominator: Failing to find the common denominator can lead to incorrect results.
• Not rewriting the fractions with the common denominator: Failing to rewrite the fractions with the common denominator can lead to incorrect results.
• Not adding the numerators: Failing to add the numerators can lead to incorrect results.
• Not simplifying the result: Failing to simplify the result can lead to incorrect results.

Real-World Applications

Adding fractions with different denominators has several real-world applications. Here are a few:

• Cooking: When cooking, you may need to add fractions of ingredients to a recipe. For example, you may need to add $\frac{1}{4}$ cup of sugar and $\frac{1}{2}$ cup of flour to a recipe.
• Science: In science, you may need to add fractions of measurements to a experiment. For example, you may need to add $\frac{1}{2}$ liter of water and $\frac{1}{4}$ liter of acid to a experiment.
• Finance: In finance, you may need to add fractions of investments to a portfolio. For example, you may need to add $\frac{1}{4}$ of a stock and $\frac{1}{2}$ of a bond to a portfolio.

Final Thoughts

Q: What is the common denominator?

A: The common denominator is the least common multiple (LCM) of the denominators of two or more fractions. It is the smallest number that both denominators can divide into evenly.

Q: How do I find the common denominator?

A: To find the common denominator, you need to factorize both expressions and find the product of the highest power of each factor that appears in either expression.

Q: What if the denominators are not factorable?

A: If the denominators are not factorable, you can use the prime factorization method to find the LCM.

Q: Can I add fractions with different denominators without finding the common denominator?

A: No, you cannot add fractions with different denominators without finding the common denominator. Failing to find the common denominator can lead to incorrect results.

Q: How do I rewrite the fractions with the common denominator?

A: To rewrite the fractions with the common denominator, you need to multiply the numerator and denominator of each fraction by the necessary factors to obtain the common denominator.

Q: Can I add the numerators without rewriting the fractions with the common denominator?

A: No, you cannot add the numerators without rewriting the fractions with the common denominator. Failing to rewrite the fractions with the common denominator can lead to incorrect results.

Q: How do I simplify the result?

A: To simplify the result, you need to factor out the greatest common factor (GCF) of the numerator.

Q: Can I simplify the result without factoring out the GCF?

A: No, you cannot simplify the result without factoring out the GCF. Failing to factor out the GCF can lead to incorrect results.

Q: What are some common mistakes to avoid when adding fractions with different denominators?

A: Some common mistakes to avoid when adding fractions with different denominators include:

• Not finding the common denominator
• Not rewriting the fractions with the common denominator
• Not adding the numerators
• Not simplifying the result

Q: What are some real-world applications of adding fractions with different denominators?

A: Some real-world applications of adding fractions with different denominators include:

• Cooking
• Science
• Finance

Q: How can I practice adding fractions with different denominators?

A: You can practice adding fractions with different denominators by using online resources, such as math worksheets and practice problems. You can also practice by working on real-world problems, such as cooking and science experiments.

Q: What are some tips for adding fractions with different denominators?

A: Some tips for adding fractions with different denominators include:

• Finding the common denominator
• Rewriting the fractions with the common denominator
• Adding the numerators
• Simplifying the result

Conclusion

Adding fractions with different denominators can be challenging, but with a clear understanding of the concept and a step-by-step approach, you can easily add fractions with different denominators and simplify the result. Remember to find the common denominator, rewrite the fractions with the common denominator, add the numerators, and simplify the result. With practice and patience, you can become proficient in adding fractions with different denominators.