After how many decimal places, expansion `23/(2^(4)xx5^(3))` will terminate?

Options

A. `2`
B. `3`
C. `4`
D. `5`

Answer

To determine how many decimal places the expansion of \( \frac{23}{2^4 \times 5^3} \) will terminate, we can follow these steps:

### Step 1: Identify the Denominator
The given expression is:
\[ \frac{23}{2^4 \times 5^3} \]
Here, the denominator consists of powers of 2 and 5.

### Step 2: Equalize the Powers of 2 and 5
For a decimal expansion to terminate, the denominator must be of the form \( 2^m \times 5^m \), where \( m \) is a non-negative integer. In our case, we have:
- Power of 2: 4
- Power of 5: 3

To equalize the powers, we can multiply the numerator and denominator by \( 5^{(4-3)} = 5^1 \):
\[ \frac{23 \times 5}{2^4 \times 5^3 \times 5^1} = \frac{115}{2^4 \times 5^4} \]

### Step 3: Simplify the Expression
Now, our expression is:
\[ \frac{115}{2^4 \times 5^4} \]
This can be rewritten as:
\[ \frac{115}{(2 \times 5)^4} = \frac{115}{10^4} \]

### Step 4: Calculate the Decimal Value
Now we can calculate:
\[ \frac{115}{10^4} = 0.0115 \]
This shows that the decimal expansion is \( 0.0115 \).

### Step 5: Determine the Number of Decimal Places
The decimal \( 0.0115 \) has 4 decimal places.

### Conclusion
Thus, the expansion of \( \frac{23}{2^4 \times 5^3} \) will terminate after **4 decimal places**.

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