Simplify The Expression: $\frac{7}{a} + \frac{5}{b}$

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Introduction

In mathematics, simplifying expressions is a crucial skill that helps us to solve problems more efficiently. When dealing with fractions, we often need to add or subtract them, which can be a challenging task. In this article, we will focus on simplifying the expression $\frac{7}{a} + \frac{5}{b}$, which is a common problem in algebra and mathematics.

Understanding the Problem

The given expression is a sum of two fractions, $\frac{7}{a}$ and $\frac{5}{b}$. To simplify this expression, we need to find a common denominator, which is the least common multiple (LCM) of the denominators $a$ and $b$. Once we have the common denominator, we can add the two fractions together.

Finding the Common Denominator

To find the common denominator, we need to find the LCM of $a$ and $b$. The LCM of two numbers is the smallest number that is a multiple of both numbers. For example, the LCM of 6 and 8 is 24, because 24 is the smallest number that is a multiple of both 6 and 8.

In this case, we can find the LCM of $a$ and $b$ by listing the multiples of each number and finding the smallest number that appears in both lists. However, this method can be time-consuming and may not be practical for large numbers.

A more efficient way to find the LCM is to use the prime factorization method. This method involves breaking down each number into its prime factors and then multiplying the highest power of each prime factor together.

Prime Factorization Method

To find the LCM of $a$ and $b$ using the prime factorization method, we need to break down each number into its prime factors. We can do this by dividing each number by its prime factors and then multiplying the results together.

For example, let's say we want to find the LCM of 12 and 15. We can break down each number into its prime factors as follows:

• 12 = 2 × 2 × 3
• 15 = 3 × 5

To find the LCM, we multiply the highest power of each prime factor together:

• LCM = 2 × 2 × 3 × 5 = 60

Therefore, the LCM of 12 and 15 is 60.

Simplifying the Expression

Now that we have found the common denominator, we can simplify the expression $\frac{7}{a} + \frac{5}{b}$ by adding the two fractions together.

$\frac{7}{a} + \frac{5}{b} = \frac{7b}{ab} + \frac{5a}{ab}$

We can combine the two fractions by adding the numerators together:

$\frac{7b}{ab} + \frac{5a}{ab} = \frac{7b + 5a}{ab}$

Therefore, the simplified expression is $\frac{7b + 5a}{ab}$.

Conclusion

Simplifying expressions is an essential skill in mathematics, and it requires a good understanding of fractions and algebra. In this article, we have focused on simplifying the expression $\frac{7}{a} + \frac{5}{b}$, which is a common problem in algebra and mathematics.

We have used the prime factorization method to find the common denominator, which is the LCM of the denominators $a$ and $b$. Once we have the common denominator, we can add the two fractions together and simplify the expression.

Example Problems

Here are some example problems that you can try to practice simplifying expressions:

• $\frac{3}{x} + \frac{2}{y}$
• $\frac{4}{z} + \frac{6}{w}$
• $\frac{9}{p} + \frac{7}{q}$

Tips and Tricks

Here are some tips and tricks that you can use to simplify expressions:

• Always find the common denominator before adding or subtracting fractions.
• Use the prime factorization method to find the LCM of two numbers.
• Combine the numerators and denominators of two fractions by adding or subtracting them.
• Simplify the expression by canceling out any common factors in the numerator and denominator.

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Introduction

In our previous article, we discussed how to simplify the expression $\frac{7}{a} + \frac{5}{b}$. We used the prime factorization method to find the common denominator, which is the least common multiple (LCM) of the denominators $a$ and $b$. In this article, we will answer some frequently asked questions about simplifying expressions.

Q&A

Q: What is the common denominator of two fractions?

A: The common denominator of two fractions is the least common multiple (LCM) of the denominators. To find the LCM, we can use the prime factorization method or list the multiples of each number and find the smallest number that appears in both lists.

Q: How do I find the LCM of two numbers using the prime factorization method?

A: To find the LCM of two numbers using the prime factorization method, we need to break down each number into its prime factors. We can do this by dividing each number by its prime factors and then multiplying the results together.

For example, let's say we want to find the LCM of 12 and 15. We can break down each number into its prime factors as follows:

• 12 = 2 × 2 × 3
• 15 = 3 × 5

To find the LCM, we multiply the highest power of each prime factor together:

• LCM = 2 × 2 × 3 × 5 = 60

Therefore, the LCM of 12 and 15 is 60.

Q: How do I simplify an expression with two fractions?

A: To simplify an expression with two fractions, we need to find a common denominator, which is the LCM of the denominators. Once we have the common denominator, we can add the two fractions together.

For example, let's say we want to simplify the expression $\frac{7}{a} + \frac{5}{b}$. We can find the common denominator by using the prime factorization method or listing the multiples of each number and finding the smallest number that appears in both lists.

Once we have the common denominator, we can add the two fractions together:

$\frac{7}{a} + \frac{5}{b} = \frac{7b}{ab} + \frac{5a}{ab}$

We can combine the two fractions by adding the numerators together:

$\frac{7b}{ab} + \frac{5a}{ab} = \frac{7b + 5a}{ab}$

Therefore, the simplified expression is $\frac{7b + 5a}{ab}$.

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

A: Some common mistakes to avoid when simplifying expressions include:

• Not finding the common denominator before adding or subtracting fractions.
• Not using the prime factorization method to find the LCM of two numbers.
• Not combining the numerators and denominators of two fractions by adding or subtracting them.
• Not simplifying the expression by canceling out any common factors in the numerator and denominator.

By avoiding these common mistakes, you can simplify expressions more efficiently and accurately.

Q: How can I practice simplifying expressions?

A: You can practice simplifying expressions by working on example problems, such as:

• $\frac{3}{x} + \frac{2}{y}$
• $\frac{4}{z} + \frac{6}{w}$
• $\frac{9}{p} + \frac{7}{q}$

You can also try simplifying expressions with different denominators, such as:

• $\frac{7}{a} + \frac{5}{b}$
• $\frac{3}{x} + \frac{4}{y}$
• $\frac{2}{z} + \frac{6}{w}$

By practicing simplifying expressions, you can become more confident and proficient in this skill.

Conclusion

Simplifying expressions is an essential skill in mathematics, and it requires a good understanding of fractions and algebra. In this article, we have answered some frequently asked questions about simplifying expressions, including how to find the common denominator, how to simplify an expression with two fractions, and how to avoid common mistakes. By following these tips and practicing simplifying expressions, you can become more confident and proficient in this skill.