To multiply the fractions \( \frac{3}{5} \), \( \frac{2}{5} \), and \( \frac{4}{9} \), you multiply the numerators together and the denominators together.
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Multiply the numerators: \[ 3 \times 2 \times 4 = 24 \]
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Multiply the denominators: \[ 5 \times 5 \times 9 = 225 \]
Putting it all together, you get: \[ \frac{24}{225} \]
Now, to simplify \( \frac{24}{225} \), we can find the greatest common divisor (GCD) of 24 and 225. The prime factorization of 24 is \( 2^3 \times 3 \), and the prime factorization of 225 is \( 3^2 \times 5^2 \). The GCD is \( 3 \).
Now, divide both the numerator and the denominator by 3: \[ \frac{24 \div 3}{225 \div 3} = \frac{8}{75} \]
So, the simplified result is: \[ \frac{8}{75} \]