Find The Quotient And Simplify.$\[ \frac{m^2-n^2}{m+n} \div \frac{m}{m^2+n M} = \square \\](Simplify Your Answer.)

Introduction

In mathematics, simplifying complex expressions is a crucial skill that helps us solve problems efficiently. One such expression is the quotient of two algebraic fractions, which can be simplified using various techniques. In this article, we will explore how to find the quotient and simplify the given expression: $\frac{m^2-n^2}{m+n} \div \frac{m}{m^2+n m} = \square$. We will break down the solution into manageable steps, making it easy to understand and follow.

Understanding the Expression

The given expression involves two algebraic fractions: $\frac{m^2-n^2}{m+n}$ and $\frac{m}{m^2+n m}$. To simplify the expression, we need to first understand the properties of these fractions. The first fraction can be simplified using the difference of squares formula, which states that $a^2 - b^2 = (a + b)(a - b)$. Applying this formula, we get:

$\frac{m^2-n^2}{m+n} = \frac{(m+n)(m-n)}{m+n} = m - n$

Simplifying the Second Fraction

The second fraction, $\frac{m}{m^2+n m}$, can be simplified by factoring out the common term $m$ from the denominator:

$\frac{m}{m^2+n m} = \frac{m}{m(m+n)} = \frac{1}{m+n}$

Finding the Quotient

Now that we have simplified both fractions, we can find the quotient by dividing the first fraction by the second fraction:

$\frac{m - n}{\frac{1}{m+n}} = (m - n)(m + n)$

Simplifying the Result

The resulting expression, $(m - n)(m + n)$, can be simplified using the difference of squares formula again:

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

Conclusion

In this article, we have simplified the given expression $\frac{m^2-n^2}{m+n} \div \frac{m}{m^2+n m} = \square$ using various techniques. We first simplified the first fraction using the difference of squares formula, then simplified the second fraction by factoring out the common term $m$ from the denominator. Finally, we found the quotient by dividing the first fraction by the second fraction and simplified the resulting expression using the difference of squares formula again. The final answer is $\boxed{m^2 - n^2}$.

Frequently Asked Questions

• Q: What is the difference of squares formula? A: The difference of squares formula states that $a^2 - b^2 = (a + b)(a - b)$.
• Q: How do I simplify the expression $\frac{m^2-n^2}{m+n} \div \frac{m}{m^2+n m}$? A: To simplify the expression, first simplify the first fraction using the difference of squares formula, then simplify the second fraction by factoring out the common term $m$ from the denominator. Finally, find the quotient by dividing the first fraction by the second fraction and simplify the resulting expression using the difference of squares formula again.
• Q: What is the final answer to the expression $\frac{m^2-n^2}{m+n} \div \frac{m}{m^2+n m}$? A: The final answer is $\boxed{m^2 - n^2}$.

Step-by-Step Solution

1. Simplify the first fraction using the difference of squares formula: $\frac{m^2-n^2}{m+n} = \frac{(m+n)(m-n)}{m+n} = m - n$
2. Simplify the second fraction by factoring out the common term $m$ from the denominator: $\frac{m}{m^2+n m} = \frac{m}{m(m+n)} = \frac{1}{m+n}$
3. Find the quotient by dividing the first fraction by the second fraction: $\frac{m - n}{\frac{1}{m+n}} = (m - n)(m + n)$
4. Simplify the resulting expression using the difference of squares formula again: $(m - n)(m + n) = m^2 - n^2$

Additional Resources

• For more information on simplifying algebraic expressions, visit the Khan Academy website.
• For practice problems and exercises, visit the Mathway website.
• For a comprehensive guide to algebra, visit the Algebra.com website.

Final Answer

The final answer to the expression $\frac{m^2-n^2}{m+n} \div \frac{m}{m^2+n m}$ is $\boxed{m^2 - n^2}$.

Introduction

Simplifying algebraic expressions can be a challenging task, especially when dealing with complex expressions. In our previous article, we explored how to simplify the expression $\frac{m^2-n^2}{m+n} \div \frac{m}{m^2+n m} = \square$. In this article, we will answer some of the most frequently asked questions related to simplifying algebraic expressions.

Q&A

Q: What is the difference of squares formula?

A: The difference of squares formula states that $a^2 - b^2 = (a + b)(a - b)$. This formula can be used to simplify expressions that involve the difference of two squares.

Q: How do I simplify the expression $\frac{m^2-n^2}{m+n} \div \frac{m}{m^2+n m}$?

A: To simplify the expression, first simplify the first fraction using the difference of squares formula, then simplify the second fraction by factoring out the common term $m$ from the denominator. Finally, find the quotient by dividing the first fraction by the second fraction and simplify the resulting expression using the difference of squares formula again.

Q: What is the final answer to the expression $\frac{m^2-n^2}{m+n} \div \frac{m}{m^2+n m}$?

A: The final answer is $\boxed{m^2 - n^2}$.

Q: How do I simplify expressions that involve the sum of two squares?

A: To simplify expressions that involve the sum of two squares, you can use the formula $a^2 + b^2 = (a + bi)(a - bi)$, where $i$ is the imaginary unit.

Q: What is the difference between simplifying and factoring?

A: Simplifying an expression involves reducing it to its simplest form, while factoring an expression involves expressing it as a product of simpler expressions.

Q: How do I simplify expressions that involve fractions?

A: To simplify expressions that involve fractions, you can use the following steps:

1. Simplify the numerator and denominator separately.
2. Cancel out any common factors between the numerator and denominator.
3. Simplify the resulting expression.

Q: What is the final answer to the expression $\frac{2x^2+5x-3}{x^2-4x+3}$?

A: To simplify the expression, first simplify the numerator and denominator separately. Then, cancel out any common factors between the numerator and denominator. Finally, simplify the resulting expression.

Step-by-Step Solution

1. Simplify the numerator and denominator separately: $\frac{2x^2+5x-3}{x^2-4x+3} = \frac{(2x-1)(x+3)}{(x-3)(x-1)}$
2. Cancel out any common factors between the numerator and denominator: $\frac{(2x-1)(x+3)}{(x-3)(x-1)} = \frac{2x-1}{x-3}$
3. Simplify the resulting expression: $\frac{2x-1}{x-3} = \frac{2(x-1/2)}{x-3}$

Additional Resources

• For more information on simplifying algebraic expressions, visit the Khan Academy website.
• For practice problems and exercises, visit the Mathway website.
• For a comprehensive guide to algebra, visit the Algebra.com website.

Final Answer

The final answer to the expression $\frac{2x^2+5x-3}{x^2-4x+3}$ is $\boxed{\frac{2(x-1/2)}{x-3}}$.