multiplying Algebraic Expressions
1.) a² - 2ab + b²               a - 1
    ___________   ×     ____________
        a² - 1                           a - b

To multiply the given algebraic expressions, we need to factor each expression where possible, multiply the numerators together, and multiply the denominators together. Then, we simplify by canceling common factors.

### Given Expressions:

\[
\frac{a^2 - 2ab + b^2}{a^2 - 1} \times \frac{a - 1}{a - b}
\]

### Step 1: Factor Each Expression

1. **Numerator of the First Expression:**
   \[
   a^2 - 2ab + b^2 = (a - b)^2
   \]
   (Perfect square trinomial)

2. **Denominator of the First Expression:**
   \[
   a^2 - 1 = (a - 1)(a + 1)
   \]
   (Difference of squares)

3. **Numerator of the Second Expression:**
   \[
   a - 1
   \]
   (Already factored)

4. **Denominator of the Second Expression:**
   \[
   a - b
   \]
   (Already factored)

### Step 2: Rewrite the Expressions

After factoring, we rewrite the expressions as:

\[
\frac{(a - b)^2}{(a - 1)(a + 1)} \times \frac{a - 1}{a - b}
\]

### Step 3: Multiply the Numerators and Denominators

\[
\frac{(a - b)^2 \cdot (a - 1)}{(a - 1)(a + 1)(a - b)}
\]

### Step 4: Simplify by Canceling Common Factors

- \(a - 1\) in the numerator and denominator.
- \(a - b\) in the numerator and denominator.

The expression reduces to:

\[
\frac{a - b}{a + 1}
\]

### Final Answer

\[
\frac{a - b}{a + 1}
\]

Would you like more details on any step, or do you have any questions?

### Related Questions to Explore:
1. How do you simplify expressions with exponents and variables?
2. What are the rules for factoring polynomials?
3. How do you identify and apply the difference of squares in algebraic expressions?
4. What techniques are used for simplifying rational expressions?
5. How can multiplying and dividing rational expressions be applied in real-world problems?

**Tip:** When multiplying or dividing algebraic fractions, always factor expressions first to identify and cancel common factors.

Learn how to multiply algebraic expressions (a² - 2ab + b²)/(a² - 1) × (a - 1)/(a - b) step-by-step with a detailed solution. This problem involves factoring, simplifying rational expressions, and applying the difference of squares formula in algebra. Suitable for high school students.

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