Simplify (a^2+ab+b^2)/(a^3-b^3)

Solution:

Step 1:

Recognize that both the numerator and the denominator are expressions involving cubes. Apply the formula for the difference of cubes: $a^3 - b^3 = (a - b)(a^2 + ab + b^2)$, where in this case $a$ and $b$ remain the same. Thus, we rewrite the expression as:

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

Step 2:

Identify and eliminate the common factor $a^2 + ab + b^2$ from both the numerator and the denominator.

Step 2.1:

Remove the shared factor by simplification:

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

Step 2.2:

Express the simplified form of the fraction:

$$\frac{1}{a - b}$$

Knowledge Notes:

The problem involves simplifying a fraction where the numerator and the denominator have algebraic expressions. The key concepts used in this problem are:

1. Difference of Cubes: This is a special factoring formula that states for any two terms $a$ and $b$, the expression $a^3 - b^3$ can be factored into $(a - b)(a^2 + ab + b^2)$. It's a useful tool for factoring expressions where each term is a cube.

2. Common Factor: In algebra, if both the numerator and the denominator of a fraction have a common factor, it can be canceled out. This is based on the property that a fraction $\frac{a}{b}$ is equivalent to $\frac{a \div c}{b \div c}$, where $c$ is a common factor of both $a$ and $b$.

3. Simplification: The process of simplifying an algebraic expression involves reducing it to its simplest form. This often means factoring expressions, canceling common factors, and combining like terms.

In this problem, the simplification process is straightforward because the numerator is a part of the factored form of the denominator. Once the common factor is identified, it is canceled out, leaving a simpler expression.