Simplify The Expression: ( 2 M + 1 2 M − 1 − 2 M − 1 2 M + 1 ) ÷ 4 M 10 M − 5 \left(\frac{2m+1}{2m-1} - \frac{2m-1}{2m+1}\right) \div \frac{4m}{10m-5} ( 2 M − 1 2 M + 1 ​ − 2 M + 1 2 M − 1 ​ ) ÷ 10 M − 5 4 M ​

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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 a complex expression involving fractions and variables. The given expression is (2m+12m−1−2m−12m+1)÷4m10m−5\left(\frac{2m+1}{2m-1} - \frac{2m-1}{2m+1}\right) \div \frac{4m}{10m-5}. Our goal is to simplify this expression step by step, using various algebraic techniques.

Understanding the Expression

Before we start simplifying the expression, let's break it down and understand its components. The expression consists of two main parts: the numerator and the denominator. The numerator is a difference of two fractions, while the denominator is a single fraction. To simplify the expression, we need to focus on the numerator first.

Simplifying the Numerator

The numerator is given by 2m+12m−1−2m−12m+1\frac{2m+1}{2m-1} - \frac{2m-1}{2m+1}. To simplify this expression, we can start by finding a common denominator for the two fractions. The common denominator is (2m−1)(2m+1)(2m-1)(2m+1).

import sympy as sp

m = sp.symbols('m')



numerator = (2m + 1)/(2m - 1) - (2m - 1)/(2m + 1)



simplified_numerator = sp.simplify(numerator)
print(simplified_numerator)

Simplifying the Denominator

The denominator is given by 4m10m−5\frac{4m}{10m-5}. To simplify this expression, we can start by factoring the denominator. We can factor out a common factor of 55 from the denominator.

# Define the denominator
denominator = 4*m/(10*m - 5)


simplified_denominator = sp.simplify(denominator)
print(simplified_denominator)

Combining the Simplified Numerator and Denominator

Now that we have simplified the numerator and denominator, we can combine them to get the final simplified expression.

# Define the final expression
final_expression = simplified_numerator / simplified_denominator


simplified_final_expression = sp.simplify(final_expression)
print(simplified_final_expression)

Conclusion

In this article, we simplified a complex expression involving fractions and variables. We started by breaking down the expression into its components and then simplified the numerator and denominator separately. Finally, we combined the simplified numerator and denominator to get the final simplified expression. The simplified expression is 25m−5\frac{2}{5m-5}.

Final Answer

The final answer is 25m−5\boxed{\frac{2}{5m-5}}.

Step-by-Step Solution

Here is the step-by-step solution to the problem:

  1. Break down the expression into its components.
  2. Simplify the numerator by finding a common denominator and combining the fractions.
  3. Simplify the denominator by factoring out a common factor.
  4. Combine the simplified numerator and denominator to get the final simplified expression.

Tips and Tricks

  • When simplifying expressions, it's essential to focus on one part at a time.
  • Use algebraic techniques such as finding common denominators and factoring to simplify expressions.
  • Combine the simplified numerator and denominator to get the final simplified expression.

Common Mistakes

  • Failing to find a common denominator when simplifying fractions.
  • Not factoring out common factors when simplifying expressions.
  • Not combining the simplified numerator and denominator to get the final simplified expression.

Real-World Applications

  • Simplifying expressions is a crucial skill in mathematics, and it has many real-world applications.
  • In physics, simplifying expressions is used to solve complex problems involving motion and energy.
  • In engineering, simplifying expressions is used to design and optimize complex systems.

Further Reading

  • For more information on simplifying expressions, check out the following resources:
  • Khan Academy: Simplifying Expressions
  • Mathway: Simplifying Expressions
  • Wolfram Alpha: Simplifying Expressions

References

Introduction

In our previous article, we simplified a complex expression involving fractions and variables. In this article, we will provide a Q&A guide to algebraic manipulation, focusing on simplifying expressions. We will answer common questions and provide tips and tricks to help you master the art of simplifying expressions.

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

A: The first step in simplifying an expression is to break it down into its components. Identify the numerator and denominator, and then simplify each part separately.

Q: How do I simplify a fraction with a variable in the denominator?

A: To simplify a fraction with a variable in the denominator, you can start by factoring out a common factor from the denominator. This will help you simplify the fraction and make it easier to work with.

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

A: Simplifying an expression involves reducing it to its simplest form, while solving an equation involves finding the value of the variable that makes the equation true.

Q: Can I simplify an expression with multiple variables?

A: Yes, you can simplify an expression with multiple variables. However, you need to be careful when simplifying expressions with multiple variables, as it can be easy to make mistakes.

Q: How do I know when an expression is simplified?

A: An expression is simplified when it cannot be reduced further. This means that there are no more common factors to factor out, and the expression is in its simplest form.

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

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

  • Failing to find a common denominator when simplifying fractions
  • Not factoring out common factors when simplifying expressions
  • Not combining the simplified numerator and denominator to get the final simplified expression

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 real-world applications of simplifying expressions?

A: Simplifying expressions has many real-world applications, including:

  • Physics: Simplifying expressions is used to solve complex problems involving motion and energy.
  • Engineering: Simplifying expressions is used to design and optimize complex systems.
  • Computer Science: Simplifying expressions is used in algorithms and data structures.

Q: Can I use technology to simplify expressions?

A: Yes, you can use technology to simplify expressions. Many calculators and computer algebra systems, such as Wolfram Alpha and Mathematica, can simplify expressions for you.

Q: How can I check my work when simplifying expressions?

A: You can check your work by plugging the simplified expression back into the original equation and verifying that it is true.

Q: What are some tips and tricks for simplifying expressions?

A: Some tips and tricks for simplifying expressions include:

  • Start by simplifying the numerator and denominator separately
  • Use algebraic techniques such as finding common denominators and factoring to simplify expressions
  • Combine the simplified numerator and denominator to get the final simplified expression

Conclusion

Simplifying expressions is a crucial skill in mathematics, and it has many real-world applications. By following the tips and tricks outlined in this article, you can master the art of simplifying expressions and become a proficient mathematician.

Final Answer

The final answer is 25m−5\boxed{\frac{2}{5m-5}}.

Step-by-Step Solution

Here is the step-by-step solution to the problem:

  1. Break down the expression into its components.
  2. Simplify the numerator by finding a common denominator and combining the fractions.
  3. Simplify the denominator by factoring out a common factor.
  4. Combine the simplified numerator and denominator to get the final simplified expression.

Further Reading

  • For more information on simplifying expressions, check out the following resources:
  • Khan Academy: Simplifying Expressions
  • Mathway: Simplifying Expressions
  • Wolfram Alpha: Simplifying Expressions

References