Simplify The Expression:$\[ \frac{a^3-b^3}{a^2-b^2} \times \frac{2a^2-ab-3a^2}{5a^2+5ab+5b} \\]

Introduction

Algebraic expressions can be complex and daunting, but with the right techniques and strategies, they can be simplified and factored to reveal their underlying structure. In this article, we will explore the process of simplifying the given expression, which involves factoring and canceling common terms. We will break down the expression into manageable parts, identify common factors, and use algebraic properties to simplify the expression.

Understanding the Expression

The given expression is a product of two fractions:

$$\frac{a^3-b^3}{a^2-b^2} \times \frac{2a^2-ab-3a^2}{5a^2+5ab+5b}$$

To simplify this expression, we need to factor the numerator and denominator of each fraction and then cancel out any common factors.

Factoring the Numerator and Denominator

Let's start by factoring the numerator and denominator of the first fraction:

$$\frac{a^3-b^3}{a^2-b^2}$$

The numerator can be factored using the difference of cubes formula:

$$a^3-b^3 = (a-b)(a^2+ab+b^2)$$

The denominator can be factored as:

$$a^2-b^2 = (a-b)(a+b)$$

Now, we can cancel out the common factor (a-b) from the numerator and denominator:

$$\frac{(a-b)(a^2+ab+b^2)}{(a-b)(a+b)} = \frac{a^2+ab+b^2}{a+b}$$

Factoring the Second Fraction

Now, let's factor the numerator and denominator of the second fraction:

$$\frac{2a^2-ab-3a^2}{5a^2+5ab+5b}$$

The numerator can be factored as:

$$2a^2-ab-3a^2 = -a^2-ab$$

The denominator can be factored as:

$$5a^2+5ab+5b = 5(a^2+ab+b)$$

Simplifying the Expression

Now that we have factored the numerator and denominator of each fraction, we can simplify the expression by canceling out any common factors.

$$\frac{a^2+ab+b^2}{a+b} \times \frac{-a^2-ab}{5(a^2+ab+b)}$$

We can cancel out the common factor (a+b) from the first fraction and the common factor (a^2+ab+b) from the second fraction:

$$\frac{a^2+ab+b^2}{a+b} \times \frac{-a^2-ab}{5(a^2+ab+b)} = \frac{-a^2-ab}{5}$$

Conclusion

In this article, we have simplified the given expression by factoring and canceling common terms. We have broken down the expression into manageable parts, identified common factors, and used algebraic properties to simplify the expression. The final simplified expression is:

$$\frac{-a^2-ab}{5}$$

This expression can be further simplified by factoring out a common term:

$$\frac{-a(a+b)}{5}$$

This is the final simplified expression.

Final Answer

The final answer is $\boxed{\frac{-a(a+b)}{5}}$.

Frequently Asked Questions

• Q: What is the difference of cubes formula? A: The difference of cubes formula is: $a^3-b^3 = (a-b)(a^2+ab+b^2)$
• Q: How do I factor the numerator and denominator of a fraction? A: To factor the numerator and denominator of a fraction, look for common factors and use algebraic properties to simplify the expression.
• Q: What is the final simplified expression? A: The final simplified expression is $\frac{-a(a+b)}{5}$.

Further Reading

• Algebraic Expressions: A Comprehensive Guide
• Factoring and Simplifying Algebraic Expressions
• Algebraic Properties: A Review
  

Introduction

Algebraic expressions can be complex and daunting, but with the right techniques and strategies, they can be simplified and factored to reveal their underlying structure. In this article, we will answer some of the most frequently asked questions about algebraic expressions, covering topics such as factoring, simplifying, and algebraic properties.

Q&A

Q: What is the difference of cubes formula?

A: The difference of cubes formula is: $a^3-b^3 = (a-b)(a^2+ab+b^2)$

Q: How do I factor the numerator and denominator of a fraction?

A: To factor the numerator and denominator of a fraction, look for common factors and use algebraic properties to simplify the expression. For example, if you have a fraction with a numerator of $a^3-b^3$ and a denominator of $a^2-b^2$, you can factor the numerator using the difference of cubes formula and the denominator using the difference of squares formula.

Q: What is the difference of squares formula?

A: The difference of squares formula is: $a^2-b^2 = (a-b)(a+b)$

Q: How do I simplify an algebraic expression?

A: To simplify an algebraic expression, look for common factors and use algebraic properties to simplify the expression. For example, if you have an expression of $\frac{a^2+ab+b^2}{a+b}$, you can simplify it by canceling out the common factor (a+b) from the numerator and denominator.

Q: What is the final simplified expression for the given problem?

A: The final simplified expression for the given problem is $\frac{-a(a+b)}{5}$.

Q: What are some common algebraic properties?

A: Some common algebraic properties include:

• The commutative property of addition: $a+b = b+a$
• The commutative property of multiplication: $a \cdot b = b \cdot a$
• The associative property of addition: $(a+b)+c = a+(b+c)$
• The associative property of multiplication: $(a \cdot b) \cdot c = a \cdot (b \cdot c)$
• The distributive property: $a(b+c) = ab+ac$

Q: How do I use algebraic properties to simplify an expression?

A: To use algebraic properties to simplify an expression, look for opportunities to apply the properties. For example, if you have an expression of $a(b+c)$, you can use the distributive property to simplify it to $ab+ac$.

Q: What is the importance of algebraic expressions in real-life applications?

A: Algebraic expressions are used in a wide range of real-life applications, including:

• Physics: Algebraic expressions are used to describe the motion of objects and the behavior of physical systems.
• Engineering: Algebraic expressions are used to design and optimize systems, such as electrical circuits and mechanical systems.
• Economics: Algebraic expressions are used to model economic systems and make predictions about future economic trends.
• Computer Science: Algebraic expressions are used to develop algorithms and solve problems in computer science.

Conclusion

In this article, we have answered some of the most frequently asked questions about algebraic expressions, covering topics such as factoring, simplifying, and algebraic properties. We have also discussed the importance of algebraic expressions in real-life applications. By understanding and applying these concepts, you can simplify and factor algebraic expressions with ease.

Final Answer

The final answer is $\boxed{\frac{-a(a+b)}{5}}$.

Further Reading

• Algebraic Expressions: A Comprehensive Guide
• Factoring and Simplifying Algebraic Expressions
• Algebraic Properties: A Review
• Real-Life Applications of Algebraic Expressions

Frequently Asked Questions

• Q: What is the difference of cubes formula? A: The difference of cubes formula is: $a^3-b^3 = (a-b)(a^2+ab+b^2)$
• Q: How do I factor the numerator and denominator of a fraction? A: To factor the numerator and denominator of a fraction, look for common factors and use algebraic properties to simplify the expression.
• Q: What is the final simplified expression for the given problem? A: The final simplified expression for the given problem is $\frac{-a(a+b)}{5}$.

Related Articles

• Algebraic Expressions: A Comprehensive Guide
• Factoring and Simplifying Algebraic Expressions
• Algebraic Properties: A Review
• Real-Life Applications of Algebraic Expressions