Simplify (a^2x-a^2y)/(a^2-ay^2)

Solution:

Step 1: Simplify the Numerator

Step 1.1: Extract the Common Factor from the Numerator

Extract $a^2$ from both terms in the numerator.

$$\frac{a^2(x) - a^2(y)}{a^2 - ay^2}$$

Step 1.2: Combine the Factored Terms

Combine the terms factored by $a^2$.

$$\frac{a^2(x - y)}{a^2 - ay^2}$$

Step 2: Simplify the Denominator

Step 2.1: Extract the Common Factor from the Denominator

Extract $a$ from both terms in the denominator.

$$\frac{a^2(x - y)}{a(a - y^2)}$$

Step 3: Cancel the Common Factors

Step 3.1: Factor Out the Common Term from the Numerator

Factor $a$ from the numerator expression.

$$\frac{a(a(x - y))}{a(a - y^2)}$$

Step 3.2: Cancel Out the Common Terms

Cancel out the common $a$ factor from the numerator and denominator.

$$\frac{\cancel{a}(a(x - y))}{\cancel{a}(a - y^2)}$$

Step 3.3: Final Simplified Expression

Write down the simplified expression.

$$\frac{a(x - y)}{a - y^2}$$

Knowledge Notes:

To simplify the given algebraic fraction, we need to apply the following knowledge points:

1. Factoring: This is the process of breaking down an expression into its multiplicative components. In this problem, we factored $a^2$ out of the numerator and $a$ out of the denominator.

2. Common Factor: A common factor is a factor that is shared by two or more terms. In this problem, $a^2$ is a common factor in the numerator, and $a$ is a common factor in the denominator.

3. Cancellation: When a factor appears in both the numerator and the denominator of a fraction, it can be cancelled out. This is because dividing by a number is the same as multiplying by its reciprocal. In this problem, we cancelled the common factor $a$.

4. Simplifying Expressions: The process of simplifying involves reducing an expression to its simplest form. This often involves factoring and cancelling, as well as combining like terms.

By applying these principles, we were able to simplify the given algebraic fraction to its simplest form.