The decimal expansion of `(23)/(2^(3) xx 5^(2))` will terminate after how many places of decimal?

Options

A. 2
B. 4
C. 3
D. 1

Answer

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

### Step 1: Identify the Denominator
The denominator is given as \(2^3 \times 5^2\).

### Step 2: Rewrite the Denominator
We can express the denominator in terms of powers of 10. Since \(10 = 2 \times 5\), we can combine the powers of 2 and 5:
\[ 2^3 \times 5^2 = 2^2 \times 5^2 \times 2^1 = (2 \times 5)^2 \times 2^1 = 10^2 \times 2^1 = 100 \times 2 = 200 \]

### Step 3: Rewrite the Fraction
Now, we can rewrite the fraction:
\[ \frac{23}{2^3 \times 5^2} = \frac{23}{200} \]

### Step 4: Multiply Numerator and Denominator
To make the denominator a power of 10, we can multiply both the numerator and the denominator by \(5\) (since we have one extra factor of 2 in the denominator):
\[ \frac{23 \times 5}{200 \times 5} = \frac{115}{1000} \]

### Step 5: Identify the Decimal Expansion
Now, we can express \(\frac{115}{1000}\) as a decimal:
\[ \frac{115}{1000} = 0.115 \]

### Step 6: Count the Decimal Places
The decimal \(0.115\) has three decimal places.

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

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