Question

Find the least common denominator for the following pair of rational expressions.
         StartFraction 25 Over 12 m EndFraction
and StartFraction 5 Over 12 m minus 20 EndFraction

Answer 1

To find the least common denominator (LCD) for the rational expressions \( \frac{25}{12m} \) and \( \frac{5}{12m - 20} \), we need to analyze the denominators of both fractions.

1. **Identify the Denominators**:
- The first expression has a denominator of \( 12m \).
- The second expression has a denominator of \( 12m - 20 \).

2. **Factor the Denominators**:
- The first denominator \( 12m \) is already simplified.
- We can factor the second denominator:
\[
12m - 20 = 4(3m - 5)
\]
So, the second denominator becomes \( 4(3m - 5) \).

3. **Determine the LCD**:
- The LCD must include each factor from the denominators without repetition.
- From \( 12m \), the factors are \( 12 \) and \( m \).
- From \( 4(3m - 5) \), the factors are \( 4 \) and \( (3m - 5) \).
- The least common multiple of \( 12 \) and \( 4 \) is \( 12 \).
- Therefore, the LCD will be:
\[
\text{LCD} = 12 \cdot m \cdot (3m - 5)
\]

4. **Final Expression for the LCD**:
- The least common denominator is:
\[
\text{LCD} = 12m(3m - 5)
\]

So, the least common denominator for the given rational expressions is \( 12m(3m - 5) \).