Rationalise the denominator of each the following
`(i)(2)/(sqrt(3))" "(ii)(1)/(3sqrt(5))" "(iii)(1)/(sqrt(8))" "(iv)(sqrt(2)+1)/(sqrt(3))`

Explanation

To rationalize the denominators of the given expressions, we will follow a systematic approach for each part. Here’s the step-by-step solution:

### (i) Rationalizing \( \frac{2}{\sqrt{3}} \)

1. **Multiply numerator and denominator by \( \sqrt{3} \)**:
\[ \frac{2}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{2\sqrt{3}}{3} \]

**Final answer**: \( \frac{2\sqrt{3}}{3} \)

### (ii) Rationalizing \( \frac{1}{3\sqrt{5}} \)

1. **Multiply numerator and denominator by \( \sqrt{5} \)**:
\[ \frac{1}{3\sqrt{5}} \times \frac{\sqrt{5}}{\sqrt{5}} = \frac{\sqrt{5}}{3 \cdot 5} = \frac{\sqrt{5}}{15} \]

**Final answer**: \( \frac{\sqrt{5}}{15} \)

### (iii) Rationalizing \( \frac{1}{\sqrt{8}} \)

1. **Multiply numerator and denominator by \( \sqrt{8} \)**:
\[ \frac{1}{\sqrt{8}} \times \frac{\sqrt{8}}{\sqrt{8}} = \frac{\sqrt{8}}{8} \]

2. **Simplify \( \sqrt{8} \)**:
\[ \sqrt{8} = 2\sqrt{2} \quad \text{(since \( 8 = 4 \times 2 \))} \]
So,
\[ \frac{\sqrt{8}}{8} = \frac{2\sqrt{2}}{8} = \frac{\sqrt{2}}{4} \]

**Final answer**: \( \frac{\sqrt{2}}{4} \)

### (iv) Rationalizing \( \frac{\sqrt{2} + 1}{\sqrt{3}} \)

1. **Multiply numerator and denominator by \( \sqrt{3} \)**:
\[ \frac{\sqrt{2} + 1}{\sqrt{3}} \times \frac{\sqrt{3}}{\sqrt{3}} = \frac{(\sqrt{2} + 1)\sqrt{3}}{3} \]

2. **Distribute in the numerator**:
\[ = \frac{\sqrt{6} + \sqrt{3}}{3} \]

**Final answer**: \( \frac{\sqrt{6} + \sqrt{3}}{3} \)

### Summary of Answers:
1. \( \frac{2\sqrt{3}}{3} \)
2. \( \frac{\sqrt{5}}{15} \)
3. \( \frac{\sqrt{2}}{4} \)
4. \( \frac{\sqrt{6} + \sqrt{3}}{3} \)