To multiply the fractions \( \frac{2}{3} \), \( \frac{2}{7} \), and \( \frac{9}{11} \), you multiply the numerators together and the denominators together:
\[ \frac{2}{3} \times \frac{2}{7} \times \frac{9}{11} = \frac{2 \times 2 \times 9}{3 \times 7 \times 11} \]
Calculating the numerator:
\[ 2 \times 2 = 4 \] \[ 4 \times 9 = 36 \]
Calculating the denominator:
\[ 3 \times 7 = 21 \] \[ 21 \times 11 = 231 \]
Now combine the results:
\[ \frac{36}{231} \]
Next, we simplify \( \frac{36}{231} \) if possible. We can find the greatest common divisor (GCD) of 36 and 231.
The factors of 36 are \( 1, 2, 3, 4, 6, 9, 12, 18, 36 \) The factors of 231 are \( 1, 3, 7, 11, 21, 33, 77, 231 \)
The greatest common factor is 3.
Now we divide both the numerator and denominator by 3:
\[ \frac{36 \div 3}{231 \div 3} = \frac{12}{77} \]
Thus, the final simplified result is:
\[ \frac{12}{77} \]