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:
- The denominator of the first fraction is \( 9(x-2) \).
- The denominator of the second fraction is \( 9 \).
- The denominator of the third fraction is \( x-2 \).
Now, we break these denominators down into their prime factors:
- \( 9(x-2) = 3^2(x-2) \)
- \( 9 = 3^2 \)
- \( x-2 = (x-2) \)
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:
\[ \text{LCD} = 9(x-2) \]