Find the least common denominator.

StartFraction 7 Over x squared minus 81 EndFraction7x2−81

and

StartFraction 2 Over x left parenthesis x minus 9 right parenthesis EndFraction2x(x−9)

To find the least common denominator (LCD) of the fractions:

\[
\frac{7}{x^2 - 81} \quad \text{and} \quad \frac{2}{x(x - 9)},
\]

we need to factorize the denominators and then determine the LCD.

### Step 1: Factorize the Denominators

1. **Factorize \(x^2 - 81\):**

   This is a difference of squares:

   \[
   x^2 - 81 = (x - 9)(x + 9).
   \]

2. **Factorize \(x(x - 9)\):**

   This is already factored.

   \[
   x(x - 9).
   \]

### Step 2: Determine the Least Common Denominator

To find the LCD, we take the highest power of each unique factor:

- Factors from \((x - 9)(x + 9)\): \( (x - 9) \) and \( (x + 9) \).
- Factors from \(x(x - 9)\): \( x \) and \( (x - 9) \).

So, the unique factors are \(x\), \( (x - 9) \), and \( (x + 9) \).

### LCD:

\[
\text{LCD} = x(x - 9)(x + 9).
\]

### Final Answer:

The least common denominator is:

\[
\boxed{x(x - 9)(x + 9)}.
\]

Would you like more details or have any questions?

### Related Questions:

1. How do you add fractions with different denominators?
2. What is the process for simplifying complex fractions?
3. How do you factor polynomials involving squares and cubes?
4. What are common strategies for finding the greatest common factor (GCF)?
5. How can the least common multiple (LCM) be applied in other contexts?

**Tip:** Always factorize denominators completely to easily find the LCD when working with fractions.

Find the least common denominator. 7/(x^2 - 81) and 2/[x(x - 9)]

Discover how to find the least common denominator (LCD) for the rational expressions 7/(x^2 - 81) and 2/[x(x - 9)]. This guide covers factoring techniques like the difference of squares, factoring polynomials, and simplifying expressions, suitable for Grades 9-10 students.

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