Express   0.3528   in the form `(p)/( 2 ^(m) 5 ^(n))` and write the values of p, m and n.

Answer

To express \(0.3528\) in the form \(\frac{p}{2^m \cdot 5^n}\), we will follow these steps:

### Step 1: Convert the Decimal to a Fraction
We start by converting \(0.3528\) into a fraction. Since there are four decimal places, we can write:
\[ 0.3528 = \frac{3528}{10000} \]

### Step 2: Simplify the Fraction
Next, we simplify \(\frac{3528}{10000}\). We need to find the greatest common divisor (GCD) of \(3528\) and \(10000\).

1. **Finding the GCD**:
- The prime factorization of \(10000\) is \(10^4 = (2 \cdot 5)^4 = 2^4 \cdot 5^4\).
- We will factor \(3528\):
- Divide \(3528\) by \(2\): \(3528 \div 2 = 1764\)
- Divide \(1764\) by \(2\): \(1764 \div 2 = 882\)
- Divide \(882\) by \(2\): \(882 \div 2 = 441\)
- Now, \(441\) is not divisible by \(2\), so we divide by \(3\): \(441 \div 3 = 147\)
- Divide \(147\) by \(3\): \(147 \div 3 = 49\)
- Finally, \(49\) is \(7^2\).

Therefore, the prime factorization of \(3528\) is:
\[ 3528 = 2^3 \cdot 3^2 \cdot 7^2 \]

2. **Finding the GCD**:
- The GCD of \(3528\) and \(10000\) involves the lowest powers of the common prime factors:
- The only common factor is \(2\), and the lowest power is \(2^3\).
- Thus, GCD is \(2^3 = 8\).

3. **Simplifying**:
- Now we can simplify \(\frac{3528}{10000}\) by dividing both the numerator and the denominator by \(8\):
\[ \frac{3528 \div 8}{10000 \div 8} = \frac{441}{1250} \]

### Step 3: Express the Denominator in Terms of \(2\) and \(5\)
Next, we express \(1250\) in terms of \(2\) and \(5\):
- The prime factorization of \(1250\) is:
\[ 1250 = 125 \cdot 10 = 5^3 \cdot (2 \cdot 5) = 2^1 \cdot 5^4 \]

### Step 4: Write in the Required Form
Now we can express \(\frac{441}{1250}\) as:
\[ \frac{441}{1250} = \frac{441}{2^1 \cdot 5^4} \]
This matches the required form \(\frac{p}{2^m \cdot 5^n}\).

### Step 5: Identify \(p\), \(m\), and \(n\)
From our expression:
- \(p = 441\)
- \(m = 1\)
- \(n = 4\)

### Final Answer
Thus, we have:
\[ p = 441, \quad m = 1, \quad n = 4 \]