Without actually performing the long division, write the decimal expansion of `11725/(2^3 times 5^4)`

Answer

To find the decimal expansion of \( \frac{11725}{2^3 \times 5^4} \) without performing long division, we can follow these steps:

### Step 1: Identify the denominator
The denominator is given as \( 2^3 \times 5^4 \). We can calculate this value first.

\[ 2^3 = 8 \quad \text{and} \quad 5^4 = 625 \]
So,

\[ 2^3 \times 5^4 = 8 \times 625 \]

### Step 2: Calculate \( 8 \times 625 \)
Now, we calculate \( 8 \times 625 \):

\[ 8 \times 625 = 5000 \]

### Step 3: Rewrite the expression
Now we can rewrite the original expression:

\[ \frac{11725}{2^3 \times 5^4} = \frac{11725}{5000} \]

### Step 4: Simplify the fraction
To simplify \( \frac{11725}{5000} \), we can multiply both the numerator and the denominator by 2 to eliminate the fraction without changing its value:

\[ \frac{11725 \times 2}{5000 \times 2} = \frac{23450}{10000} \]

### Step 5: Convert to decimal
Now, we can convert \( \frac{23450}{10000} \) to decimal form. This is done by moving the decimal point 4 places to the left (since the denominator is \( 10000 \)):

\[ 23450 \div 10000 = 2.345 \]

### Final Answer
Thus, the decimal expansion of \( \frac{11725}{2^3 \times 5^4} \) is:

\[ \boxed{2.345} \]