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}
]
