To solve the expression ( \frac{8}{81} \cdot 9:32 ), we first need to recognize that the colon (:) represents a ratio or division, which can be expressed as a fraction.
The expression ( 9:32 ) can be rewritten as ( \frac{9}{32} ).
Now, we can rewrite the original expression:
[
\frac{8}{81} \cdot \frac{9}{32}
]
Next, we multiply the numerators and the denominators:
[
= \frac{8 \cdot 9}{81 \cdot 32}
]
Calculating the numerator:
[
8 \cdot 9 = 72
]
Calculating the denominator:
[
81 \cdot 32 = 2592
]
So, we have:
[
\frac{72}{2592}
]
Now, we can simplify this fraction. To do that, we find the greatest common divisor (GCD) of 72 and 2592.
Calculating the GCD:
- 72 can be factored into ( 2^3 \cdot 3^2 )
- 2592 can be factored into ( 2^5 \cdot 3^4 )
The GCD is ( 2^3 \cdot 3^2 = 72 ).
Now, we divide both the numerator and denominator by the GCD:
[
\frac{72 \div 72}{2592 \div 72} = \frac{1}{36}
]
Thus, the simplified result of the original expression ( \frac{8}{81} \cdot 9:32 ) is:
[
\frac{1}{36}
]
