2/3 x 9/38 x 8/27
To multiply the fractions \( \frac{2}{3} \), \( \frac{9}{38} \), and \( \frac{8}{27} \), you can follow these steps:
Multiply the numerators: \[ 2 \times 9 \times 8 = 144 \]
Multiply the denominators: \[ 3 \times 38 \times 27 = 3066 \]
So, the expression becomes: \[ \frac{144}{3066} \]
Next, we should simplify this fraction. First, we can find the greatest common divisor (GCD) of 144 and 3066.
The prime factorization of 144 is: \[ 144 = 2^4 \times 3^2 \]
Finding the prime factorization of 3066:
- Divide by 2: \( 3066 \div 2 = 1533 \)
- Divide by 3: \( 1533 \div 3 = 511 \)
- 511 is prime.
Thus, the prime factorization of 3066 is: \[ 3066 = 2^1 \times 3^1 \times 511 \]
Now, the GCD is:
- For 2, the minimum exponent is \( 1 \).
- For 3, the minimum exponent is \( 1 \).
So, the GCD is: \[ 2^1 \times 3^1 = 6 \]
Now, divide both the numerator and the denominator by the GCD (6): \[ \frac{144 \div 6}{3066 \div 6} = \frac{24}{511} \]
Thus, the simplified result is: \[ \frac{24}{511} \]
