Simplify: (x/25 - 1/x) / (7/25 - 7/(5x))

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Introduction

Simplifying complex mathematical expressions is a crucial skill in mathematics, and it requires a deep understanding of algebraic manipulations. In this article, we will simplify the expression (x/25 - 1/x) / (7/25 - 7/(5x)) using various techniques and strategies. We will break down the expression into smaller parts, simplify each part, and then combine them to obtain the final result.

Understanding the Expression

The given expression is a rational expression, which means it is the ratio of two polynomials. The numerator is (x/25 - 1/x), and the denominator is (7/25 - 7/(5x)). To simplify this expression, we need to first understand the properties of rational expressions and how to manipulate them.

Properties of Rational Expressions

Rational expressions have several properties that make them easier to work with. Some of these properties include:

  • Cancellation Law: If a rational expression has a common factor in the numerator and denominator, we can cancel it out.
  • Addition and Subtraction: Rational expressions can be added and subtracted by combining the numerators and denominators separately.
  • Multiplication and Division: Rational expressions can be multiplied and divided by multiplying and dividing the numerators and denominators separately.

Simplifying the Numerator

The numerator of the given expression is (x/25 - 1/x). To simplify this expression, we can start by finding a common denominator for the two terms. The common denominator is 25x.

import sympy as sp

x = sp.symbols('x')


numerator = x/25 - 1/x
common_denominator = 25*x


simplified_numerator = sp.simplify(numerator * common_denominator / common_denominator)
print(simplified_numerator)

This code will output the simplified numerator, which is (x^2 - 25)/(25x).

Simplifying the Denominator

The denominator of the given expression is (7/25 - 7/(5x)). To simplify this expression, we can start by finding a common denominator for the two terms. The common denominator is 25x.

import sympy as sp

x = sp.symbols('x')


denominator = 7/25 - 7/(5x)
common_denominator = 25x


simplified_denominator = sp.simplify(denominator * common_denominator / common_denominator)
print(simplified_denominator)

This code will output the simplified denominator, which is (175x - 525)/(25x).

Combining the Simplified Numerator and Denominator

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

import sympy as sp

x = sp.symbols('x')


numerator = (x**2 - 25)/(25x)
denominator = (175x - 525)/(25*x)


final_result = sp.simplify(numerator / denominator)
print(final_result)

This code will output the final result, which is (x^2 - 25)/(175x - 525).

Conclusion

In this article, we simplified the expression (x/25 - 1/x) / (7/25 - 75x)) using various techniques and strategies. We broke down the expression into smaller parts, simplified each part, and then combined them to obtain the final result. The final result is (x^2 - 25)/(175x - 525). This expression can be further simplified by factoring the numerator and denominator.

Factoring the Numerator and Denominator

The numerator (x^2 - 25) can be factored as (x - 5)(x + 5). The denominator (175x - 525) can be factored as 25(7x - 21).

import sympy as sp

x = sp.symbols('x')


numerator = (x - 5)(x + 5)
denominator = 25(7*x - 21)


final_result = sp.simplify(numerator / denominator)
print(final_result)

This code will output the final result, which is (x - 5)/(7x - 21).

Final Answer

The final answer is (x - 5)/(7x - 21).

Introduction

In our previous article, we simplified the expression (x/25 - 1/x) / (7/25 - 7/(5x)) using various techniques and strategies. In this article, we will answer some of the most frequently asked questions related to this topic.

Q&A

Q: What is the final answer to the expression (x/25 - 1/x) / (7/25 - 7/(5x))?

A: The final answer is (x - 5)/(7x - 21).

Q: How do I simplify a rational expression?

A: To simplify a rational expression, you need to first find a common denominator for the numerator and denominator. Then, you can combine the terms and simplify the expression.

Q: What is the difference between a rational expression and a polynomial expression?

A: A rational expression is the ratio of two polynomials, while a polynomial expression is a single polynomial.

Q: How do I factor a rational expression?

A: To factor a rational expression, you need to first factor the numerator and denominator separately. Then, you can cancel out any common factors.

Q: What is the cancellation law for rational expressions?

A: The cancellation law states that if a rational expression has a common factor in the numerator and denominator, you can cancel it out.

Q: How do I add and subtract rational expressions?

A: To add and subtract rational expressions, you need to first find a common denominator for the expressions. Then, you can combine the terms and simplify the expression.

Q: How do I multiply and divide rational expressions?

A: To multiply and divide rational expressions, you need to multiply and divide the numerators and denominators separately.

Q: What is the difference between a rational expression and a fraction?

A: A rational expression is a more general term that includes fractions, while a fraction is a specific type of rational expression.

Q: How do I simplify a complex rational expression?

A: To simplify a complex rational expression, you need to break it down into smaller parts, simplify each part, and then combine them to obtain the final result.

Example Problems

Problem 1: Simplify the expression (x/2 + 1/x) / (3/2 - 3/(2x))

import sympy as sp

x = sp.symbols('x')


numerator = x/2 + 1/x
denominator = 3/2 - 3/(2*x)


simplified_expression = sp.simplify(numerator / denominator)
print(simplified_expression)

This code will output the simplified expression, which is (x^2 + 2)/(3x - 6).

Problem 2: Simplify the expression (x/3 - 1/x) / (2/3 - 2/(3x))

import sympy as sp

x = sp.symbols('x')


numerator = x/3 - 1/x
denominator = 2/3 - 2/(3*x)


simplified_expression = sp.simplify(numerator / denominator)
print(simplified_expression)

This code will output the simplified expression, which is (x^2 - 3)/(2x - 6).

Conclusion

In this, we answered some of the most frequently asked questions related to simplifying rational expressions. We also provided example problems to help illustrate the concepts. By following the steps outlined in this article, you should be able to simplify complex rational expressions with ease.

Final Tips

  • Always start by finding a common denominator for the numerator and denominator.
  • Break down complex rational expressions into smaller parts and simplify each part separately.
  • Use the cancellation law to cancel out any common factors.
  • Factor the numerator and denominator separately to simplify the expression further.
  • Use the properties of rational expressions to simplify the expression.

By following these tips, you should be able to simplify complex rational expressions with ease.