https://file.aiquickdraw.com/mathbot/file/1727056196603d92odiwp.jpg

The expression you are being asked to simplify is:

\[
\frac{r^2 - 81}{r^2 - 18r + 81}
\]

### Step-by-step Simplification:

1. **Factor the numerator:**
   The numerator \(r^2 - 81\) is a difference of squares:
   \[
   r^2 - 81 = (r - 9)(r + 9)
   \]

2. **Factor the denominator:**
   The denominator \(r^2 - 18r + 81\) is a perfect square trinomial:
   \[
   r^2 - 18r + 81 = (r - 9)(r - 9) = (r - 9)^2
   \]

3. **Simplify the expression:**
   Now substitute the factored forms into the original expression:
   \[
   \frac{(r - 9)(r + 9)}{(r - 9)^2}
   \]
   Cancel out the common factor \((r - 9)\) from the numerator and denominator:
   \[
   \frac{r + 9}{r - 9}
   \]

The simplified expression is:

\[
\frac{r + 9}{r - 9}
\]

This is the simplest form of the given expression after removing the factors of 1.

---

Do you have any questions or would you like further clarification? 

Here are 5 related questions to explore this concept further:
1. How do you identify a difference of squares in an algebraic expression?
2. What is the process for factoring a perfect square trinomial?
3. When simplifying rational expressions, what conditions must be met for factors to be canceled?
4. Can this simplification be applied to all values of \(r\), or are there any restrictions?
5. How would the simplification change if the expression had a different structure, such as \(r^2 + 18r + 81\)?

**Tip:** Always check for common factors in both the numerator and denominator before canceling terms to avoid mistakes.

Simplify the expression by removing factors of 1: (r^2 - 81) / (r^2 - 18r + 81)

Learn how to simplify the rational expression (r^2 - 81) / (r^2 - 18r + 81) by factoring a difference of squares and a perfect square trinomial. This step-by-step solution is perfect for students in Grades 9-12, covering essential factoring techniques.

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