Simplify The Expression:${ 1 - N \left(\frac{3}{m-n} - \frac{2}{m+n}\right) }$

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Introduction

In mathematics, simplifying expressions is a crucial skill that helps us solve problems efficiently. It involves rewriting complex expressions in a simpler form, making it easier to understand and work with. In this article, we will focus on simplifying the given expression: 1−n(3m−n−2m+n)1 - n \left(\frac{3}{m-n} - \frac{2}{m+n}\right). We will break down the expression into smaller parts, simplify each part, and then combine them to get the final result.

Understanding the Expression

The given expression is a combination of fractions and variables. To simplify it, we need to understand the structure of the expression and identify the common terms. The expression can be broken down into three main parts:

  • The first part is the constant term, which is 1.
  • The second part is the term involving the variable n, which is −n(3m−n)-n \left(\frac{3}{m-n}\right).
  • The third part is the term involving the variable n, which is +n(2m+n)+n \left(\frac{2}{m+n}\right).

Simplifying the Expression

To simplify the expression, we will start by simplifying each part separately and then combine them.

Simplifying the First Part

The first part of the expression is the constant term, which is 1. This term does not involve any variables, so it cannot be simplified further.

Simplifying the Second Part

The second part of the expression involves the variable n and the fraction 3m−n\frac{3}{m-n}. To simplify this part, we can start by simplifying the fraction.

import sympy as sp

m, n = sp.symbols('m n')



fraction = 3 / (m - n)
simplified_fraction = sp.simplify(fraction)
print(simplified_fraction)

The simplified fraction is 3m−n\frac{3}{m-n}. Now, we can multiply this fraction by the variable n to get the second part of the expression.

# Multiply the simplified fraction by n
second_part = -n * simplified_fraction
print(second_part)

The second part of the expression is −3nm−n-\frac{3n}{m-n}.

Simplifying the Third Part

The third part of the expression involves the variable n and the fraction 2m+n\frac{2}{m+n}. To simplify this part, we can start by simplifying the fraction.

# Simplify the fraction
fraction = 2 / (m + n)
simplified_fraction = sp.simplify(fraction)
print(simplified_fraction)

The simplified fraction is 2m+n\frac{2}{m+n}. Now, we can multiply this fraction by the variable n to get the third part of the expression.

# Multiply the simplified fraction by n
third_part = n * simplified_fraction
print(third_part)

The third part of the expression is 2nm+n\frac{2n}{m+n}.

Combining the Parts

Now that we have simplified each part of the expression, we can combine them to get the final result.

# Combine the parts
expression = 1 + second_part + third_part
print(expression)

The final result is 1−3nm−n+2nm+n1 - \frac{3n}{m-n} + \frac{2n}{m+n}.

Final Answer

The final answer is 1−3nm−n+2nm+n\boxed{1 - \frac{3n}{m-n} + \frac{2n}{m+n}}.

Conclusion

Simplifying expressions is an essential skill in mathematics that helps us solve problems efficiently. By breaking down complex expressions into smaller parts, simplifying each part, and then combining them, we can get the final result. In this article, we simplified the given expression 1−n(3m−n−2m+n)1 - n \left(\frac{3}{m-n} - \frac{2}{m+n}\right) and arrived at the final answer 1−3nm−n+2nm+n\boxed{1 - \frac{3n}{m-n} + \frac{2n}{m+n}}.

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Introduction

In our previous article, we simplified the expression 1−n(3m−n−2m+n)1 - n \left(\frac{3}{m-n} - \frac{2}{m+n}\right) and arrived at the final answer 1−3nm−n+2nm+n\boxed{1 - \frac{3n}{m-n} + \frac{2n}{m+n}}. In this article, we will answer some frequently asked questions related to simplifying expressions.

Q&A

Q: What is the purpose of simplifying expressions?

A: The purpose of simplifying expressions is to rewrite complex expressions in a simpler form, making it easier to understand and work with. Simplifying expressions helps us to:

  • Identify patterns and relationships between variables
  • Solve problems more efficiently
  • Understand the underlying structure of the expression

Q: How do I simplify a complex expression?

A: To simplify a complex expression, follow these steps:

  1. Break down the expression into smaller parts
  2. Simplify each part separately
  3. Combine the simplified parts to get the final result

Q: What are some common techniques for simplifying expressions?

A: Some common techniques for simplifying expressions include:

  • Factoring: breaking down an expression into simpler factors
  • Canceling: canceling out common terms in an expression
  • Combining like terms: combining terms with the same variable and coefficient

Q: How do I simplify an expression with fractions?

A: To simplify an expression with fractions, follow these steps:

  1. Simplify each fraction separately
  2. Combine the simplified fractions to get the final result
  3. Use the least common multiple (LCM) to combine fractions with different denominators

Q: What is the difference between simplifying and solving an expression?

A: Simplifying an expression involves rewriting it in a simpler form, while solving an expression involves finding the value of the expression. For example, simplifying the expression 2x+32x + 3 results in 2x+32x + 3, while solving the expression 2x+3=52x + 3 = 5 results in x=1x = 1.

Q: Can I use a calculator to simplify expressions?

A: Yes, you can use a calculator to simplify expressions. However, it's essential to understand the underlying math and be able to simplify expressions manually. This will help you to:

  • Identify patterns and relationships between variables
  • Solve problems more efficiently
  • Understand the underlying structure of the expression

Conclusion

Simplifying expressions is an essential skill in mathematics that helps us solve problems efficiently. By understanding the purpose of simplifying expressions, using common techniques, and practicing regularly, you can become proficient in simplifying expressions. In this article, we answered some frequently asked questions related to simplifying expressions and provided tips and techniques for simplifying expressions.

Final Tips

  • Practice simplifying expressions regularly to become proficient
  • Use common techniques such as factoring, canceling, and combining like terms
  • Understand the underlying math and be able to simplify expressions manually
  • Use a calculator to simplify expressions, but also understand the underlying math

By following these tips and techniques, you can become proficient in simplifying expressions and solve problems more efficiently.