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Without actually performing the long div...

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

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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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