8/81 • 9:32

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:

\[ \frac{1}{36} \]