After how many decimal places will the decimal expansion of the rational number `(47)/(2^(3) 5^(2))` terminate?

Answer

To determine how many decimal places the decimal expansion of the rational number \( \frac{47}{2^3 \cdot 5^2} \) will terminate, we can follow these steps:

### Step 1: Identify the form of the rational number
The rational number is given as \( \frac{47}{2^3 \cdot 5^2} \). We need to check if the denominator can be expressed in the form \( 2^m \cdot 5^n \).

**Hint:** A rational number has a terminating decimal expansion if its denominator (after simplification) can be expressed as a product of the prime factors 2 and 5 only.

### Step 2: Write down the prime factors of the denominator
The denominator is \( 2^3 \cdot 5^2 \). Here, we can see:
- \( m = 3 \) (the exponent of 2)
- \( n = 2 \) (the exponent of 5)

**Hint:** Identify the exponents of the prime factors of the denominator.

### Step 3: Determine the maximum of m and n
Now, we need to find the maximum of \( m \) and \( n \):
- \( m = 3 \)
- \( n = 2 \)

The maximum value is \( \max(m, n) = \max(3, 2) = 3 \).

**Hint:** The number of decimal places is determined by the larger of the two exponents.

### Step 4: Conclusion
The decimal expansion of \( \frac{47}{2^3 \cdot 5^2} \) will terminate after \( 3 \) decimal places.

**Final Answer:** The decimal expansion will terminate after **3 decimal places**.