Examine whether the following numbers are rational or irrational. Give the decimal representation of rational numbers :
`(i)sqrt(2.56)" "(ii)2sqrt(3)" "(iii)(sqrt(36))/(20)" "(iv)sqrt(8)`

Explanation

Let's examine whether the following numbers are rational or irrational and provide the decimal representation for the rational numbers.

### (i) \(\sqrt{2.56}\)

1. **Convert to Fraction**:
\[ 2.56 = \frac{256}{100} \]
2. **Take the Square Root**:
\[ \sqrt{2.56} = \sqrt{\frac{256}{100}} = \frac{\sqrt{256}}{\sqrt{100}} = \frac{16}{10} = 1.6 \]
3. **Determine Rationality**:
Since \(1.6\) can be expressed as a fraction \(\frac{16}{10}\), it is a rational number.

**Decimal Representation**: \(1.6\)

### (ii) \(2\sqrt{3}\)

1. **Analyze the Expression**:
The number \(2\sqrt{3}\) cannot be simplified to a fraction because \(\sqrt{3}\) is an irrational number.
2. **Determine Rationality**:
Since it involves an irrational number, \(2\sqrt{3}\) is also irrational.

**Decimal Representation**: Not applicable (irrational).

### (iii) \(\frac{\sqrt{36}}{20}\)

1. **Calculate the Square Root**:
\[ \sqrt{36} = 6 \]
2. **Simplify the Expression**:
\[ \frac{\sqrt{36}}{20} = \frac{6}{20} = \frac{3}{10} \]
3. **Determine Rationality**:
Since \(\frac{3}{10}\) can be expressed as a fraction, it is a rational number.

**Decimal Representation**: \(0.3\)

### (iv) \(\sqrt{8}\)

1. **Simplify the Square Root**:
\[ \sqrt{8} = \sqrt{4 \times 2} = \sqrt{4} \times \sqrt{2} = 2\sqrt{2} \]
2. **Determine Rationality**:
Since \(\sqrt{2}\) is irrational, \(2\sqrt{2}\) is also irrational.

**Decimal Representation**: Not applicable (irrational).

### Summary of Results:
- (i) \(\sqrt{2.56}\) is rational, decimal representation: \(1.6\)
- (ii) \(2\sqrt{3}\) is irrational
- (iii) \(\frac{\sqrt{36}}{20}\) is rational, decimal representation: \(0.3\)
- (iv) \(\sqrt{8}\) is irrational