Without actually performing the long division, write the decimal expansion of `(1175)/(2^(3) xx 5^(4))`

Answer

To solve the problem of finding the decimal expansion of \( \frac{1175}{2^3 \times 5^4} \) without performing long division, we can follow these steps:

### Step 1: Simplify the Denominator
First, we need to simplify the denominator \( 2^3 \times 5^4 \).

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

Now, calculate the product:

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

### Step 2: Calculate \( 8 \times 625 \)
To find \( 8 \times 625 \):

\[ 8 \times 625 = 5000 \]

Thus, the denominator becomes \( 5000 \).

### Step 3: Rewrite the Fraction
Now we can rewrite the original fraction:

\[ \frac{1175}{2^3 \times 5^4} = \frac{1175}{5000} \]

### Step 4: Simplify the Fraction
Next, we can simplify \( \frac{1175}{5000} \). We can divide both the numerator and the denominator by 25:

\[ 1175 \div 25 = 47 \quad \text{and} \quad 5000 \div 25 = 200 \]

So, we have:

\[ \frac{1175}{5000} = \frac{47}{200} \]

### Step 5: Convert to Decimal
Now, we can convert \( \frac{47}{200} \) into decimal form. We know that \( \frac{1}{200} = 0.005 \), so:

\[ \frac{47}{200} = 47 \times 0.005 = 0.235 \]

### Final Answer
Thus, the decimal expansion of \( \frac{1175}{2^3 \times 5^4} \) is:

\[ \boxed{0.235} \]

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