Question

what is the LCD of this group of fractions?

2/9(x-2) and 5/9 and 1/x-2

Answer 1

To find the least common denominator (LCD) of the fractions \( \frac{2}{9(x-2)} \), \( \frac{5}{9} \), and \( \frac{1}{x-2} \), we first identify the denominators of each fraction:

1. The denominator of the first fraction is \( 9(x-2) \).
2. The denominator of the second fraction is \( 9 \).
3. The denominator of the third fraction is \( x-2 \).

Now, we break these denominators down into their prime factors:

1. \( 9(x-2) = 3^2(x-2) \)
2. \( 9 = 3^2 \)
3. \( x-2 = (x-2) \)

Next, to find the LCD, we take the highest power of each factor present in any of the denominators:

- The highest power of \( 3 \) from our denominators is \( 3^2 \).
- The highest power of \( (x-2) \) from our denominators is \( (x-2)^1 \).

Now we combine these to get the LCD:

\[
\text{LCD} = 3^2 (x-2) = 9(x-2)
\]

Thus, the least common denominator of the given fractions is:

\[
\text{LCD} = 9(x-2)
\]