Without 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)` (iii) `(121)/(400)` (iv) `(19)/(800)` (v) `(9)/(2^(4) xx 5^(2))` (vi) `(11)/(25)`

Answer

To determine whether each of the given rational numbers is a terminating decimal without actual division, we need to express the denominators in the form of \(2^m \times 5^n\). If the denominator can be expressed in this form, then the decimal representation of the number is terminating.

Let's solve each part step by step.

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

1. **Identify the denominator**: \(2^2 \times 5^3\)
2. **Check the form**: The denominator is in the form \(2^m \times 5^n\), where \(m = 2\) and \(n = 3\). Thus, it is a terminating decimal.
3. **Calculate the decimal**:
\[ \frac{17}{2^2 \times 5^3} = \frac{17}{4 \times 125} = \frac{17}{500} \]
Now, converting \(\frac{17}{500}\) to decimal gives:
\[ 17 \div 500 = 0.034 \]

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

1. **Identify the denominator**: \(625 = 5^4\)
2. **Check the form**: The denominator can be expressed as \(5^4 \times 2^0\). Thus, it is a terminating decimal.
3. **Calculate the decimal**:
\[ \frac{24}{625} \]
Converting gives:
\[ 24 \div 625 = 0.0384 \]

### (iii) \(\frac{121}{400}\)

1. **Identify the denominator**: \(400 = 2^4 \times 5^2\)
2. **Check the form**: The denominator is in the form \(2^m \times 5^n\), where \(m = 4\) and \(n = 2\). Thus, it is a terminating decimal.
3. **Calculate the decimal**:
\[ \frac{121}{400} \]
Converting gives:
\[ 121 \div 400 = 0.3025 \]

### (iv) \(\frac{19}{800}\)

1. **Identify the denominator**: \(800 = 2^5 \times 5^2\)
2. **Check the form**: The denominator is in the form \(2^m \times 5^n\), where \(m = 5\) and \(n = 2\). Thus, it is a terminating decimal.
3. **Calculate the decimal**:
\[ \frac{19}{800} \]
Converting gives:
\[ 19 \div 800 = 0.02375 \]

### (v) \(\frac{9}{2^4 \times 5^2}\)

1. **Identify the denominator**: \(2^4 \times 5^2\)
2. **Check the form**: The denominator is in the form \(2^m \times 5^n\), where \(m = 4\) and \(n = 2\). Thus, it is a terminating decimal.
3. **Calculate the decimal**:
\[ \frac{9}{2^4 \times 5^2} = \frac{9}{16 \times 25} = \frac{9}{400} \]
Converting gives:
\[ 9 \div 400 = 0.0225 \]

### (vi) \(\frac{11}{25}\)

1. **Identify the denominator**: \(25 = 5^2\)
2. **Check the form**: The denominator can be expressed as \(5^2 \times 2^0\). Thus, it is a terminating decimal.
3. **Calculate the decimal**:
\[ \frac{11}{25} \]
Converting gives:
\[ 11 \div 25 = 0.44 \]

### Summary of Results:
1. \(\frac{17}{500} = 0.034\)
2. \(\frac{24}{625} = 0.0384\)
3. \(\frac{121}{400} = 0.3025\)
4. \(\frac{19}{800} = 0.02375\)
5. \(\frac{9}{400} = 0.0225\)
6. \(\frac{11}{25} = 0.44\)