Final answer:
The least common denominator (LCD) of the two rational expressions with denominators 6b and 9 is found by factoring the denominators, identifying the LCM of these factors, and then constructing the LCD from this LCM which is 18b.
Step-by-step explanation:
To find the least common denominator (LCD) of two rational expressions, b - ³⁶/6b and b/9, we need to identify the least common multiple (LCM) of the denominators, which in this case are 6b and 9.
Firstly, factor both denominators to find the prime factors:
Next, the LCM must include all prime factors present, raised to their highest power that occurs in any of the denominators. Here the prime factors are 2, 3, and b. Since the highest power of 3 appearing in any denominator is 32 (or 9), the LCD will be:
2 × 32 × b = 18b
This LCD allows us to combine or compare the two rational expressions without altering their values.