The decimal expansion of the rational number `(27)/(2^(2) * 5^(3))`, will terminate after how many places of decimals ?

Answer

To determine how many places the decimal expansion of the rational number \(\frac{27}{2^2 \cdot 5^3}\) will terminate, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Denominator**:
The denominator of the given rational number is \(2^2 \cdot 5^3\).

2. **Express the Denominator in the Form of Powers of 2 and 5**:
Here, we can see that:
- The power of 2 (m) is 2.
- The power of 5 (n) is 3.

3. **Determine the Maximum Power**:
We need to find the maximum of the two powers:
\[ m = 2 \quad \text{and} \quad n = 3 \]
The maximum value is:
\[ \max(m, n) = \max(2, 3) = 3 \]

4. **Conclusion**:
Since the maximum power of the denominator is 3, the decimal expansion of the rational number \(\frac{27}{2^2 \cdot 5^3}\) will terminate after 3 decimal places.

### Final Answer:
The decimal expansion of \(\frac{27}{2^2 \cdot 5^3}\) will terminate after **3 places of decimals**.