Final answer:
To find the least common denominator (LCD) for (7x^2)/((x-2)^2) and (9x)/(4x-8), identify the highest powers of each unique factor in the denominators. The LCD is 4(x-2)^2, which includes the factor (x-2) squared and the unique factor 4.
Step-by-step explanation:
The task here is to find the least common denominator (LCD) for the two given expressions, (7x^2)/((x-2)^2) and (9x)/(4x-8). The given expressions have denominators (x-2)^2 and 4(x-2), respectively. To find the LCD, we need to look for each unique factor from both denominators and take the highest power of these factors that appears in either expression.
Firstly, we factor 4(x-2) to make it easier to compare with (x-2)^2. The common factor here is (x-2). The highest power of (x-2) present in the denominators is 2, from (x-2)^2. Since the second denominator only has a single (x-2) factor, we do not need to multiply it by anything. However, the second denominator has an additional factor of 4 that the first denominator does not have.
Therefore, the LCD will be the combination of all these unique factors raised to their highest powers observed: 4(x-2)^2. This is the least common denominator for the two expressions.