After how many places will the decimal expansion of `(189)/(125)` terminate?

Answer

To determine how many places the decimal expansion of \( \frac{189}{125} \) will terminate, we can follow these steps:

### Step 1: Identify the denominator
The denominator of the fraction is \( 125 \).

### Step 2: Factor the denominator
We need to express \( 125 \) in terms of its prime factors:
\[ 125 = 5^3 \]

### Step 3: Check for factors of 2
Next, we check if there are any factors of \( 2 \) in the denominator. Since \( 125 \) can be expressed as \( 5^3 \), it does not have any factors of \( 2 \):
\[ 125 = 2^0 \times 5^3 \]
Here, \( m = 0 \) (the power of \( 2 \)) and \( n = 3 \) (the power of \( 5 \)).

### Step 4: Determine the maximum of \( m \) and \( n \)
Now we compare \( m \) and \( n \):
- \( m = 0 \)
- \( n = 3 \)

Since \( n \) is greater than \( m \), the decimal expansion will terminate after \( n \) places.

### Step 5: Conclusion
Thus, the decimal expansion of \( \frac{189}{125} \) will terminate after \( 3 \) places.

### Final Answer:
The decimal expansion of \( \frac{189}{125} \) will terminate after **3 places**.

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