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.
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Identify the Denominators:
- The first expression has a denominator of \( 12m \).
- The second expression has a denominator of \( 12m - 20 \).
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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 \( 4(3m - 5) \).
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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: \[ \text{LCD} = 12 \cdot m \cdot (3m - 5) \]
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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 \( 12m(3m - 5) \).