Simplify The Expression:${ \frac{3(m+n)-2(m-n)}{(m-n)(m+n)} }$

Introduction

Algebraic manipulation is a crucial skill in mathematics, and simplifying expressions is an essential part of it. In this article, we will focus on simplifying the given expression: $\frac{3(m+n)-2(m-n)}{(m-n)(m+n)}$. We will break down the steps involved in simplifying this expression and provide a clear understanding of the process.

Understanding the Expression

The given expression is a rational expression, which means it is the ratio of two polynomials. The numerator is $3(m+n)-2(m-n)$, and the denominator is $(m-n)(m+n)$. To simplify this expression, we need to apply the rules of algebraic manipulation, which include combining like terms, factoring, and canceling common factors.

Step 1: Expand the Numerator

To simplify the expression, we start by expanding the numerator. We can do this by applying the distributive property, which states that for any real numbers $a$, $b$, and $c$, $a(b+c) = ab + ac$. Applying this property to the numerator, we get:

$$3(m+n)-2(m-n) = 3m + 3n - 2m + 2n$$

Step 2: Combine Like Terms

Now that we have expanded the numerator, we can combine like terms. Like terms are terms that have the same variable raised to the same power. In this case, we have two terms with the variable $m$ and two terms with the variable $n$. We can combine these terms by adding or subtracting their coefficients. Applying this rule, we get:

$$3m + 3n - 2m + 2n = m + 5n$$

Step 3: Factor the Denominator

The denominator of the expression is $(m-n)(m+n)$. We can factor this expression by applying the distributive property in reverse. This means that we can rewrite the product of two binomials as the sum of two products. Applying this rule, we get:

$$(m-n)(m+n) = m^2 - n^2$$

Step 4: Simplify the Expression

Now that we have expanded the numerator and factored the denominator, we can simplify the expression. We can do this by canceling common factors between the numerator and the denominator. In this case, we have the term $m$ in the numerator and the term $m^2$ in the denominator. We can cancel these terms by dividing both the numerator and the denominator by $m$. Applying this rule, we get:

$$\frac{m + 5n}{m^2 - n^2}$$

Step 5: Final Simplification

The expression is now simplified, but we can still simplify it further by factoring the denominator. We can factor the difference of squares $m^2 - n^2$ as $(m+n)(m-n)$. Applying this rule, we get:

$$\frac{m + 5n}{(m+n)(m-n)}$$

Conclusion

In this article, we have simplified the given expression $\frac{3(m+n)-2(m-n)}{(m-n)(m+n)}$ by applying the rules of algebraic manipulation. We have expanded the numerator, combined like terms, factored the denominator, and canceled common factors. The final simplified expression is $\frac{m + 5n}{(m+n)(m-n)}$. This expression is now in its simplest form, and we can use it to solve algebraic problems.

Frequently Asked Questions

• Q: What is the purpose of simplifying expressions in algebra? A: The purpose of simplifying expressions in algebra is to make them easier to work with and to reduce the complexity of the problem.
• Q: How do I simplify a rational expression? A: To simplify a rational expression, you need to apply the rules of algebraic manipulation, which include combining like terms, factoring, and canceling common factors.
• Q: What is the difference of squares formula? A: The difference of squares formula is $a^2 - b^2 = (a+b)(a-b)$.

Final Thoughts

Simplifying expressions is an essential skill in algebra, and it requires a clear understanding of the rules of algebraic manipulation. By following the steps outlined in this article, you can simplify any rational expression and make it easier to work with. Remember to always apply the rules of algebraic manipulation in the correct order, and don't be afraid to ask for help if you get stuck. With practice and patience, you will become proficient in simplifying expressions and solving algebraic problems.

Introduction

In our previous article, we simplified the expression $\frac{3(m+n)-2(m-n)}{(m-n)(m+n)}$ by applying the rules of algebraic manipulation. In this article, we will provide a Q&A guide to help you understand the process of simplifying expressions and answer common questions related to algebraic manipulation.

Q&A Guide

Q: What is the purpose of simplifying expressions in algebra?

A: The purpose of simplifying expressions in algebra is to make them easier to work with and to reduce the complexity of the problem. Simplifying expressions can help you to:

• Solve algebraic problems more efficiently
• Identify patterns and relationships between variables
• Make predictions and conclusions based on the data

Q: How do I simplify a rational expression?

A: To simplify a rational expression, you need to apply the rules of algebraic manipulation, which include:

• Combining like terms
• Factoring
• Canceling common factors

Here's a step-by-step guide to simplifying a rational expression:

1. Expand the numerator and denominator
2. Combine like terms in the numerator and denominator
3. Factor the numerator and denominator
4. Cancel common factors between the numerator and denominator

Q: What is the difference of squares formula?

A: The difference of squares formula is $a^2 - b^2 = (a+b)(a-b)$. This formula can be used to factor the difference of squares in the denominator of a rational expression.

Q: How do I factor a quadratic expression?

A: To factor a quadratic expression, you need to find two numbers whose product is equal to the constant term and whose sum is equal to the coefficient of the linear term. Here's a step-by-step guide to factoring a quadratic expression:

1. Identify the constant term and the coefficient of the linear term
2. Find two numbers whose product is equal to the constant term and whose sum is equal to the coefficient of the linear term
3. Write the quadratic expression as a product of two binomials

Q: What is the greatest common factor (GCF) of two expressions?

A: The greatest common factor (GCF) of two expressions is the largest expression that divides both expressions without leaving a remainder. To find the GCF of two expressions, you need to:

• List the factors of each expression
• Identify the common factors between the two expressions
• Multiply the common factors to find the GCF

Q: How do I cancel common factors between the numerator and denominator?

A: To cancel common factors between the numerator and denominator, you need to:

• Identify the common factors between the numerator and denominator
• Divide both the numerator and denominator by the common factor
• Simplify the resulting expression

Q: What is the final simplified expression?

A: The final simplified expression is $\frac{m + 5n}{(m+n)(m-n)}$. This expression is now in its simplest form, and we can use it to solve algebraic problems.

Conclusion

In this article, we provided a Q&A guide to help you understand the process of simplifying expressions and answer common questions related to algebraic manipulation. By following the steps outlined in this article, you can simplify any rational expression and make it easier to work with. Remember to always apply the rules of algebraic manipulation in the correct order, and don't be afraid to ask for help if you get stuck.

Frequently Asked Questions

• Q: What is the purpose of simplifying expressions in algebra? A: The purpose of simplifying expressions in algebra is to make them easier to work with and to reduce the complexity of the problem.
• Q: How do I simplify a rational expression? A: To simplify a rational expression, you need to apply the rules of algebraic manipulation, which include combining like terms, factoring, and canceling common factors.
• Q: What is the difference of squares formula? A: The difference of squares formula is $a^2 - b^2 = (a+b)(a-b)$.

Final Thoughts

Simplifying expressions is an essential skill in algebra, and it requires a clear understanding of the rules of algebraic manipulation. By following the steps outlined in this article, you can simplify any rational expression and make it easier to work with. Remember to always apply the rules of algebraic manipulation in the correct order, and don't be afraid to ask for help if you get stuck. With practice and patience, you will become proficient in simplifying expressions and solving algebraic problems.