By actual division, show that each of the following rational numbers is a terminating decimal. Express each in the decimal form :
(i) `(17)/(2^(2) xx 5^(3))` (ii) `(24)/(625)`

Answer

To show that each of the following rational numbers is a terminating decimal, we will perform actual division for each case.

### (i) \( \frac{17}{2^2 \times 5^3} \)

1. **Calculate the denominator**:
\[ 2^2 = 4 \quad \text{and} \quad 5^3 = 125 \]
Therefore,
\[ 2^2 \times 5^3 = 4 \times 125 = 500. \]
Now we rewrite the fraction:
\[ \frac{17}{500}. \]

2. **Perform the division**:
We need to divide 17 by 500. Since 500 is greater than 17, we start by placing a decimal point:
\[ 17.000 \div 500. \]
- 500 goes into 1700 (after adding two zeros) three times (3 × 500 = 1500).
- Subtract:
\[ 1700 - 1500 = 200. \]
- Bring down another zero to make it 2000.
- 500 goes into 2000 four times (4 × 500 = 2000).
- Subtract:
\[ 2000 - 2000 = 0. \]
Thus, the division is complete.

3. **Result**:
The result of the division is:
\[ 0.034. \]

### (ii) \( \frac{24}{625} \)

1. **Perform the division**:
We need to divide 24 by 625. Since 625 is greater than 24, we place a decimal point:
\[ 24.000 \div 625. \]
- 625 goes into 2400 (after adding two zeros) three times (3 × 625 = 1875).
- Subtract:
\[ 2400 - 1875 = 525. \]
- Bring down another zero to make it 5250.
- 625 goes into 5250 eight times (8 × 625 = 5000).
- Subtract:
\[ 5250 - 5000 = 250. \]
- Bring down another zero to make it 2500.
- 625 goes into 2500 four times (4 × 625 = 2500).
- Subtract:
\[ 2500 - 2500 = 0. \]
Thus, the division is complete.

2. **Result**:
The result of the division is:
\[ 0.0384. \]

### Final Answers:
- (i) \( \frac{17}{500} = 0.034 \)
- (ii) \( \frac{24}{625} = 0.0384 \)