Simplify The Expression:${ \frac{xy}{5x^2} \times \frac{5x^2y}{x-y} }$

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Introduction

Algebraic manipulation is a crucial aspect of mathematics, and simplifying expressions is an essential skill that every student and mathematician should possess. In this article, we will focus on simplifying the given expression: xy5x2×5x2yx−y\frac{xy}{5x^2} \times \frac{5x^2y}{x-y}. We will break down the problem into manageable steps, and by the end of this article, you will have a clear understanding of how to simplify complex algebraic expressions.

Understanding the Expression

Before we dive into simplifying the expression, let's take a closer look at what we are dealing with. The given expression is a product of two fractions: xy5x2\frac{xy}{5x^2} and 5x2yx−y\frac{5x^2y}{x-y}. To simplify this expression, we need to apply the rules of algebra, which include the commutative, associative, and distributive properties.

Step 1: Simplify the First Fraction

The first fraction is xy5x2\frac{xy}{5x^2}. We can simplify this fraction by canceling out common factors in the numerator and denominator. In this case, we can cancel out the xx in the numerator with the xx in the denominator, leaving us with y5x\frac{y}{5x}.

Step 2: Simplify the Second Fraction

The second fraction is 5x2yx−y\frac{5x^2y}{x-y}. We can simplify this fraction by canceling out common factors in the numerator and denominator. In this case, we can cancel out the yy in the numerator with the yy in the denominator, leaving us with 5x2x−y\frac{5x^2}{x-y}.

Step 3: Multiply the Two Fractions

Now that we have simplified both fractions, we can multiply them together to get the final expression. To multiply fractions, we simply multiply the numerators together and the denominators together. In this case, we get:

y5x×5x2x−y=y×5x25x×(x−y)\frac{y}{5x} \times \frac{5x^2}{x-y} = \frac{y \times 5x^2}{5x \times (x-y)}

Step 4: Simplify the Final Expression

We can simplify the final expression by canceling out common factors in the numerator and denominator. In this case, we can cancel out the 5x5x in the numerator with the 5x5x in the denominator, leaving us with:

y×x2x×(x−y)=x2yx(x−y)\frac{y \times x^2}{x \times (x-y)} = \frac{x^2y}{x(x-y)}

Step 5: Factor the Denominator

The denominator of the final expression is x(x−y)x(x-y). We can factor this expression by recognizing that it is a difference of squares. In this case, we can factor the denominator as:

x(x−y)=x2−xyx(x-y) = x^2 - xy

Step 6: Simplify the Final Expression

Now that we have factored the denominator, we can simplify the final expression by canceling out common factors in the numerator and denominator. In this case, we can cancel out the xx in the numerator with the xx in the denominator, leaving us with:

x2yx2−xy=xyx−y\frac{x^2y}{x^2 - xy} = \frac{xy}{x - y}

Conclusion

In this article, we have simplified the given expression: xy5x2×5x2yx−y\frac{xy}{5x^2} \times \frac{5x^2y}{x-y}. We have broken down the problem into manageable steps, and by the end of this article, you should have a clear understanding of how to simplify complex algebraic expressions. Remember to always apply the rules of algebra, including the commutative, associative, and distributive properties, and to simplify fractions by canceling out common factors in the numerator and denominator.

Frequently Asked Questions

  • Q: What is the final simplified expression? A: The final simplified expression is xyx−y\frac{xy}{x - y}.
  • Q: How do I simplify complex algebraic expressions? A: To simplify complex algebraic expressions, you need to apply the rules of algebra, including the commutative, associative, and distributive properties, and to simplify fractions by canceling out common factors in the numerator and denominator.
  • Q: What is the difference of squares? A: The difference of squares is a mathematical expression that can be factored as the difference between two squares. In this case, we factored the denominator as x2−xyx^2 - xy.

Additional Resources

  • For more information on algebraic manipulation, please refer to the following resources:
  • Khan Academy: Algebra
  • Mathway: Algebra
  • Wolfram Alpha: Algebra

Final Thoughts

Simplifying algebraic expressions is an essential skill that every student and mathematician should possess. By following the steps outlined in this article, you should be able to simplify complex algebraic expressions with ease. Remember to always apply the rules of algebra, including the commutative, associative, and distributive properties, and to simplify fractions by canceling out common factors in the numerator and denominator. With practice and patience, you will become proficient in simplifying algebraic expressions and solving complex mathematical problems.

Introduction

Algebraic expression simplification is a crucial aspect of mathematics, and it can be a challenging task for many students and mathematicians. In our previous article, we provided a step-by-step guide on how to simplify the expression: xy5x2×5x2yx−y\frac{xy}{5x^2} \times \frac{5x^2y}{x-y}. In this article, we will provide a Q&A guide to help you better understand the concept of algebraic expression simplification.

Q&A Guide

Q: What is algebraic expression simplification?

A: Algebraic expression simplification is the process of reducing a complex algebraic expression to its simplest form by applying the rules of algebra, including the commutative, associative, and distributive properties.

Q: Why is algebraic expression simplification important?

A: Algebraic expression simplification is important because it helps to:

  • Reduce complex expressions to their simplest form
  • Make it easier to solve mathematical problems
  • Improve understanding of mathematical concepts
  • Enhance problem-solving skills

Q: What are the rules of algebra that I need to follow?

A: The rules of algebra that you need to follow are:

  • Commutative property: The order of the factors does not change the result
  • Associative property: The order in which you perform operations does not change the result
  • Distributive property: You can distribute a factor to each term in an expression

Q: How do I simplify fractions?

A: To simplify fractions, you need to cancel out common factors in the numerator and denominator. You can do this by:

  • Canceling out common factors in the numerator and denominator
  • Simplifying the fraction by dividing both the numerator and denominator by their greatest common divisor (GCD)

Q: What is the difference of squares?

A: The difference of squares is a mathematical expression that can be factored as the difference between two squares. It is written in the form: a2−b2=(a+b)(a−b)a^2 - b^2 = (a + b)(a - b)

Q: How do I factor the denominator?

A: To factor the denominator, you need to:

  • Look for common factors in the denominator
  • Factor the denominator using the difference of squares formula
  • Simplify the expression by canceling out common factors

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

A: Some common mistakes to avoid when simplifying algebraic expressions are:

  • Not canceling out common factors in the numerator and denominator
  • Not simplifying the fraction by dividing both the numerator and denominator by their GCD
  • Not factoring the denominator correctly
  • Not simplifying the expression by canceling out common factors

Q: How can I practice simplifying algebraic expressions?

A: You can practice simplifying algebraic expressions by:

  • Working on math problems and exercises
  • Using online resources and tools, such as math software and websites
  • Joining a study group or finding a study partner
  • Taking online courses or tutorials

Conclusion

Algebraic expression simplification is a crucial aspect of mathematics, and it can be a challenging task for many students and mathematicians. By following the rules of algebra, simplifying fractions, and factoring the denominator, you can simplify complex algebraic expressions with ease. Remember to avoid common mistakes and practice regularly to improve your skills.

Frequently Asked Questions

  • Q: What is the final simplified expression? A: The final simplified expression is xyx−y\frac{xy}{x - y}.
  • Q: How do I simplify complex algebraic expressions? A: To simplify complex algebraic expressions, you need to apply the rules of algebra, including the commutative, associative, and distributive properties, and to simplify fractions by canceling out common factors in the numerator and denominator.
  • Q: What is the difference of squares? A: The difference of squares is a mathematical expression that can be factored as the difference between two squares. In this case, we factored the denominator as x2−xyx^2 - xy.

Additional Resources

  • For more information on algebraic expression simplification, please refer to the following resources:
  • Khan Academy: Algebra
  • Mathway: Algebra
  • Wolfram Alpha: Algebra

Final Thoughts

Simplifying algebraic expressions is an essential skill that every student and mathematician should possess. By following the rules of algebra, simplifying fractions, and factoring the denominator, you can simplify complex algebraic expressions with ease. Remember to avoid common mistakes and practice regularly to improve your skills. With practice and patience, you will become proficient in simplifying algebraic expressions and solving complex mathematical problems.