Simplify The Expression:$\[ \frac{3}{c+7} + \frac{9}{c+7} \\]

Introduction

In mathematics, simplifying expressions is a crucial skill that helps us solve problems efficiently. When dealing with fractions, combining them can be a bit challenging, but with the right approach, it can be a breeze. In this article, we will focus on simplifying the expression $\frac{3}{c+7} + \frac{9}{c+7}$ using a step-by-step approach.

Understanding the Expression

The given expression is a sum of two fractions with the same denominator, which is $(c+7)$. To simplify this expression, we need to combine the two fractions into one. The first step is to identify the common denominator, which in this case is $(c+7)$.

Step 1: Identify the Common Denominator

The common denominator is the denominator that is common to both fractions. In this case, the common denominator is $(c+7)$. We can see that both fractions have the same denominator, which is $(c+7)$.

Step 2: Rewrite the Fractions with the Common Denominator

To combine the fractions, we need to rewrite them with the common denominator. We can do this by multiplying the numerator and denominator of each fraction by the necessary factor to make the denominator equal to the common denominator.

\frac{3}{c+7} = \frac{3(c+7)}{(c+7)(c+7)}
\frac{9}{c+7} = \frac{9(c+7)}{(c+7)(c+7)}

Step 3: Combine the Fractions

Now that we have rewritten the fractions with the common denominator, we can combine them by adding the numerators.

\frac{3(c+7)}{(c+7)(c+7)} + \frac{9(c+7)}{(c+7)(c+7)} = \frac{3(c+7) + 9(c+7)}{(c+7)(c+7)}

Step 4: Simplify the Expression

The next step is to simplify the expression by combining like terms in the numerator.

\frac{3(c+7) + 9(c+7)}{(c+7)(c+7)} = \frac{12(c+7)}{(c+7)(c+7)}

Step 5: Cancel Out the Common Factors

The final step is to cancel out the common factors in the numerator and denominator.

\frac{12(c+7)}{(c+7)(c+7)} = \frac{12}{c+7}

Conclusion

In this article, we simplified the expression $\frac{3}{c+7} + \frac{9}{c+7}$ using a step-by-step approach. We identified the common denominator, rewrote the fractions with the common denominator, combined the fractions, simplified the expression, and finally canceled out the common factors. By following these steps, we were able to simplify the expression and arrive at the final answer.

Tips and Tricks

• When dealing with fractions, always identify the common denominator before combining them.
• Rewrite the fractions with the common denominator to make it easier to combine them.
• Combine the fractions by adding the numerators.
• Simplify the expression by combining like terms in the numerator.
• Cancel out the common factors in the numerator and denominator.

Common Mistakes to Avoid

• Not identifying the common denominator before combining fractions.
• Not rewriting the fractions with the common denominator.
• Not combining the fractions by adding the numerators.
• Not simplifying the expression by combining like terms in the numerator.
• Not canceling out the common factors in the numerator and denominator.

Real-World Applications

Simplifying expressions is a crucial skill that has many real-world applications. In mathematics, it is used to solve problems in algebra, geometry, and calculus. In science, it is used to model real-world phenomena and make predictions. In engineering, it is used to design and optimize systems.

Conclusion

Introduction

In our previous article, we simplified the expression $\frac{3}{c+7} + \frac{9}{c+7}$ using a step-by-step approach. In this article, we will answer some frequently asked questions about simplifying expressions.

Q&A

Q: What is the first step in simplifying an expression?

A: The first step in simplifying an expression is to identify the common denominator. The common denominator is the denominator that is common to both fractions.

Q: How do I identify the common denominator?

A: To identify the common denominator, look for the denominator that is common to both fractions. In the expression $\frac{3}{c+7} + \frac{9}{c+7}$, the common denominator is $(c+7)$.

Q: What is the next step after identifying the common denominator?

A: After identifying the common denominator, rewrite the fractions with the common denominator. This involves multiplying the numerator and denominator of each fraction by the necessary factor to make the denominator equal to the common denominator.

Q: How do I combine the fractions?

A: To combine the fractions, add the numerators. The numerators are the numbers on top of the fractions.

Q: What is the final step in simplifying an expression?

A: The final step in simplifying an expression is to cancel out the common factors in the numerator and denominator. This involves simplifying the expression by combining like terms in the numerator and canceling out any common factors.

Q: What are some common mistakes to avoid when simplifying expressions?

A: Some common mistakes to avoid when simplifying expressions include not identifying the common denominator, not rewriting the fractions with the common denominator, not combining the fractions by adding the numerators, not simplifying the expression by combining like terms in the numerator, and not canceling out the common factors in the numerator and denominator.

Q: What are some real-world applications of simplifying expressions?

A: Simplifying expressions has many real-world applications, including solving problems in algebra, geometry, and calculus, modeling real-world phenomena, and making predictions.

Q: How can I practice simplifying expressions?

A: You can practice simplifying expressions by working through examples and exercises. Start with simple expressions and gradually move on to more complex ones.

Q: What are some tips for simplifying expressions?

A: Some tips for simplifying expressions include identifying the common denominator first, rewriting the fractions with the common denominator, combining the fractions by adding the numerators, simplifying the expression by combining like terms in the numerator, and canceling out the common factors in the numerator and denominator.

Conclusion

In conclusion, simplifying expressions is a crucial skill that has many real-world applications. By following the steps outlined in this article and practicing regularly, you can become proficient in simplifying expressions and solving problems in mathematics.

Additional Resources

• For more examples and exercises on simplifying expressions, visit our website or consult a mathematics textbook.
• For help with specific problems or questions, contact a mathematics tutor or teacher.
• For online resources and tools to help with simplifying expressions, visit our website or search online.

Final Tips

• Always identify the common denominator first.
• Rewrite the fractions with the common denominator.
• Combine the fractions by adding the numerators.
• Simplify the expression by combining like terms in the numerator.
• Cancel out the common factors in the numerator and denominator.

By following these tips and practicing regularly, you can become proficient in simplifying expressions and solving problems in mathematics.