Adding Fractions With Variables: A Step-by-Step Guide

Hey guys! Today, we're diving into the world of fractions, but with a twist – we're adding fractions that contain variables. Don't worry, it's not as scary as it sounds! We'll break it down step by step so you can confidently tackle these problems. Our specific problem is: ${\frac{5}{6c^5d^2} + \frac{2}{18c^5d^3}}$ Let's get started!

1. Finding the Least Common Denominator (LCD)

The first thing we need to do when adding fractions is to find the least common denominator (LCD). The LCD is the smallest multiple that both denominators share. In our case, the denominators are ${6c^5d^2}$ and ${18c^5d^3}$. To find the LCD, we need to consider both the numerical coefficients and the variable parts.

Numerical Coefficients

Let's start with the numbers: 6 and 18. What's the smallest number that both 6 and 18 divide into evenly? That's 18! Because 6 * 3 = 18, 18 is a multiple of 6. So, the least common multiple for the coefficients is 18.

Variable Parts

Now, let's look at the variable parts: ${c^5d^2}$ and ${c^5d^3}$. When finding the LCD for variables, we take the highest power of each variable present in either denominator.

• For ${c}$, both terms have ${c^5}$, so the LCD will also have ${c^5}$.
• For ${d}$, we have ${d^2}$ and ${d^3}$. The highest power is ${d^3}$, so the LCD will have ${d^3}$.

Combining the numerical coefficient and the variable parts, our LCD is ${18c^5d^3}$.

2. Adjusting the Fractions

Now that we have the LCD, we need to rewrite each fraction so that it has the LCD as its denominator. This means we'll be multiplying the numerator and denominator of each fraction by a suitable factor.

First Fraction: ${\frac{5}{6c^5d^2}}$

We want to change the denominator from ${6c^5d^2}$ to ${18c^5d^3}$. To do this, we need to figure out what to multiply ${6c^5d^2}$ by to get ${18c^5d^3}$.

• For the numerical coefficient, we need to multiply 6 by 3 to get 18.
• For the variable ${c}$, we already have ${c^5}$ in both denominators, so we don't need to multiply by any ${c}$.
• For the variable ${d}$, we need to multiply ${d^2}$ by ${d}$ to get ${d^3}$.

So, we need to multiply both the numerator and denominator of the first fraction by ${3d}$: ${\frac{5}{6c^5d^2} \times \frac{3d}{3d} = \frac{15d}{18c^5d^3}}$

Second Fraction: ${\frac{2}{18c^5d^3}}$

The second fraction already has the LCD as its denominator, so we don't need to change it. It remains as: ${\frac{2}{18c^5d^3}}$

3. Adding the Fractions

Now that both fractions have the same denominator, we can add them by simply adding their numerators. The denominator stays the same. ${\frac{15d}{18c^5d^3} + \frac{2}{18c^5d^3} = \frac{15d + 2}{18c^5d^3}}$

So, the sum of the two fractions is ${\frac{15d + 2}{18c^5d^3}}$.

4. Simplifying the Result (If Possible)

Finally, we need to check if we can simplify the resulting fraction. In this case, the numerator ${15d + 2}$ and the denominator ${18c^5d^3}$ don't have any common factors that we can cancel out. The expression is already in its simplest form.

Therefore, the final answer is:

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

Key Takeaways

Let's recap the steps we took to add these fractions with variables:

1. Find the Least Common Denominator (LCD): Identify the smallest multiple that both denominators share, considering both numerical coefficients and variable parts.
2. Adjust the Fractions: Multiply the numerator and denominator of each fraction by a suitable factor to get the LCD as the new denominator.
3. Add the Fractions: Add the numerators of the fractions while keeping the denominator the same.
4. Simplify the Result: Check if the resulting fraction can be simplified by canceling out common factors.

Practice Problems

To solidify your understanding, try these practice problems:

1. Add: ${\frac{3}{4x^2y} + \frac{5}{8xy^2}}$
2. Add: ${\frac{1}{2a^3b^2} + \frac{4}{6a^2b^3}}$

Remember to follow the steps we discussed, and you'll be adding fractions with variables like a pro in no time! Good luck, and have fun with it!

Conclusion

Adding fractions with variables might seem tricky at first, but with a clear understanding of the steps involved, it becomes quite manageable. By finding the LCD, adjusting the fractions, adding them, and simplifying the result, you can confidently solve these types of problems. Keep practicing, and you'll master this skill in no time. Remember, math is all about practice, practice, practice! Keep up the great work, and you'll be amazed at how far you can go! This detailed walkthrough provides a great foundation for understanding fraction addition with variables. Now you can confidently tackle similar problems. Let me know if you have any other questions, guys! Happy calculating!

I hope this article helps you grasp the concepts and provides a valuable learning experience. Feel free to reach out if you need further clarification or assistance. Always remember, practice makes perfect. The more you practice, the easier these types of problems will become. Keep exploring the world of mathematics, and you'll discover fascinating patterns and relationships. This problem serves as a building block for more advanced algebraic concepts, so mastering it now will benefit you in the long run. So, keep up the fantastic work and never stop learning! And that's how you add fractions with variables! I hope you found this helpful and easy to follow. Remember to break down complex problems into smaller, manageable steps, and you'll be able to solve anything! Keep practicing and stay curious! You've got this!