Simplify The Expression: A 3 + B 3 A 2 − A B + B 2 + A 3 − B 3 A 2 + A B + B 2 \frac{a^3+b^3}{a^2-ab+b^2}+\frac{a^3-b^3}{a^2+ab+b^2} A 2 − Ab + B 2 A 3 + B 3 + A 2 + Ab + B 2 A 3 − B 3

Mar 8, 2025 by ADMIN 189 views

$Simplify the Expression: $\frac{a^3+b^3}{a^2-ab+b^2}+\frac{a^3-b^3}{a^2+ab+b^2}$$

Introduction to the Problem

The given expression involves the sum of two rational expressions, each containing the sum or difference of cubes in the numerator and a quadratic expression in the denominator. The goal is to simplify this expression by combining the two rational expressions into a single, more manageable form.

Understanding the Structure of the Expression

To simplify the given expression, we need to first understand its structure. The expression consists of two rational expressions, each with a cubic expression in the numerator and a quadratic expression in the denominator. The first rational expression is $\frac{a^3+b^3}{a^2-ab+b^2}$ , and the second is $\frac{a^3-b^3}{a^2+ab+b^2}$ .

Factoring the Numerators and Denominators

We can start by factoring the numerators and denominators of each rational expression. The numerator of the first expression, $a^3+b^3$ , can be factored using the sum of cubes formula: $a^3+b^3 = (a+b)(a^2-ab+b^2)$ . Similarly, the numerator of the second expression, $a^3-b^3$ , can be factored using the difference of cubes formula: $a^3-b^3 = (a-b)(a^2+ab+b^2)$ .

Simplifying the Rational Expressions

Now that we have factored the numerators and denominators, we can simplify each rational expression. The first rational expression becomes $\frac{(a+b)(a^2-ab+b^2)}{a^2-ab+b^2}$ , which simplifies to $a+b$ . Similarly, the second rational expression becomes $\frac{(a-b)(a^2+ab+b^2)}{a^2+ab+b^2}$ , which simplifies to $a-b$ .

Combining the Rational Expressions

Now that we have simplified each rational expression, we can combine them by adding them together. The simplified expression is $a+b+(a-b)$ .

Final Simplification

The final step is to simplify the combined expression. We can combine like terms by adding the $a$ terms and the $b$ terms separately. This gives us $2a$ .

Conclusion

In conclusion, the given expression $\frac{a^3+b^3}{a^2-ab+b^2}+\frac{a^3-b^3}{a^2+ab+b^2}$ can be simplified by factoring the numerators and denominators, simplifying each rational expression, and combining them. The final simplified expression is $2a$ .

Step-by-Step Solution

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

Factor the numerators and denominators of each rational expression.
Simplify each rational expression by canceling out common factors.
Combine the simplified rational expressions by adding them together.
Simplify the combined expression by combining like terms.

Example Use Case

The given expression can be used to simplify more complex expressions involving the sum or difference of cubes. For example, if we have the expression $\frac{a^3+b^3+c^3}{a^2-ab+bc^2}+\frac{a^3-b^3+c^3}{a^2+ab+bc^2}$ , we can use the same steps to simplify it.

Tips and Tricks

Here are some tips and tricks to help you simplify expressions involving the sum or difference of cubes:

Use the sum of cubes formula: $a^3+b^3 = (a+b)(a^2-ab+b^2)$ .
Use the difference of cubes formula: $a^3-b^3 = (a-b)(a^2+ab+b^2)$ .
Factor the numerators and denominators of each rational expression.
Simplify each rational expression by canceling out common factors.
Combine the simplified rational expressions by adding them together.
Simplify the combined expression by combining like terms.

Common Mistakes to Avoid

Here are some common mistakes to avoid when simplifying expressions involving the sum or difference of cubes:

Not factoring the numerators and denominators of each rational expression.
Not simplifying each rational expression by canceling out common factors.
Not combining the simplified rational expressions by adding them together.
Not simplifying the combined expression by combining like terms.

Final Answer

The final answer is: $\boxed{2a}$

Introduction

In our previous article, we simplified the expression $\frac{a^3+b^3}{a^2-ab+b^2}+\frac{a^3-b^3}{a^2+ab+b^2}$ to $2a$ . However, we understand that some readers may still have questions about the problem. In this article, we will address some of the most frequently asked questions about the problem.

Q&A

Q: What is the sum of cubes formula?

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

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 numerators and denominators of each rational expression?

A: To factor the numerators and denominators of each rational expression, you can use the sum of cubes formula and the difference of cubes formula. For example, the numerator of the first expression, $a^3+b^3$ , can be factored as $(a+b)(a^2-ab+b^2)$ .

Q: How do I simplify each rational expression?

A: To simplify each rational expression, you can cancel out common factors. For example, the first rational expression becomes $\frac{(a+b)(a^2-ab+b^2)}{a^2-ab+b^2}$ , which simplifies to $a+b$ .

Q: How do I combine the simplified rational expressions?

A: To combine the simplified rational expressions, you can add them together. For example, the simplified expression is $a+b+(a-b)$ .

Q: How do I simplify the combined expression?

A: To simplify the combined expression, you can combine like terms. For example, the final simplified expression is $2a$ .

Q: What are some common mistakes to avoid when simplifying expressions involving the sum or difference of cubes?

A: Some common mistakes to avoid when simplifying expressions involving the sum or difference of cubes include not factoring the numerators and denominators of each rational expression, not simplifying each rational expression by canceling out common factors, not combining the simplified rational expressions by adding them together, and not simplifying the combined expression by combining like terms.

Q: Can I use this method to simplify more complex expressions?

A: Yes, you can use this method to simplify more complex expressions involving the sum or difference of cubes. For example, if you have the expression $\frac{a^3+b^3+c^3}{a^2-ab+bc^2}+\frac{a^3-b^3+c^3}{a^2+ab+bc^2}$ , you can use the same steps to simplify it.