Let's simplify the expression by canceling common factors.

The expression is as follows:

$\frac{x^2 - x - 6}{x^2 - 6x - 16} \cdot \frac{x + 2}{x - 8} \cdot \frac{x - 3}{x + 2} \cdot \frac{x - 3}{x - 8} \cdot \frac{x - 8}{x - 3}$

Step 1: Factor the quadratic expressions

We start by factoring both the numerator and denominator of the first fraction.

• The numerator $x^2 - x - 6$ factors as $(x - 3)(x + 2)$.
• The denominator $x^2 - 6x - 16$ factors as $(x - 8)(x + 2)$.

Now, the expression becomes:

$\frac{(x - 3)(x + 2)}{(x - 8)(x + 2)} \cdot \frac{x + 2}{x - 8} \cdot \frac{x - 3}{x + 2} \cdot \frac{x - 3}{x - 8} \cdot \frac{x - 8}{x - 3}$

Step 2: Cancel common factors

Now, we cancel out the common factors that appear in both the numerator and denominator across the entire expression:

• The factor $(x + 2)$ appears in both the numerator and denominator, so it cancels out.
• The factor $(x - 3)$ appears in both the numerator and denominator, so it cancels out.
• The factor $(x - 8)$ appears in both the numerator and denominator, so it cancels out.

After canceling all common factors, the entire expression simplifies to:

$1$

Thus, the simplified expression is:

$1$

Do you want details or have any questions?

Here are some related questions to explore this further:

1. How do you factor quadratic expressions efficiently?
2. What are some other examples of rational expressions that can be simplified?
3. How can you identify common factors in more complex expressions?
4. What is the difference between factoring and expanding expressions?
5. How can you simplify an expression involving higher-degree polynomials?

Tip: Always check for common factors in both the numerator and denominator before performing multiplications or divisions in rational expressions.