Rationalise the denominators of the following:(i) `1/(sqrt(7))` (ii) `1/(sqrt(7)-sqrt(6))` (iii) `1/(sqrt(5)+sqrt(2))` (iv) `1/(sqrt(7)-2)`

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 for each part:

### (i) Rationalize \( \frac{1}{\sqrt{7}} \)

1. **Multiply by the conjugate**: Multiply the numerator and denominator by \( \sqrt{7} \):
\[ \frac{1 \cdot \sqrt{7}}{\sqrt{7} \cdot \sqrt{7}} = \frac{\sqrt{7}}{7} \]

**Final Answer**: \( \frac{\sqrt{7}}{7} \)

### (ii) Rationalize \( \frac{1}{\sqrt{7} - \sqrt{6}} \)

1. **Multiply by the conjugate**: Multiply the numerator and denominator by \( \sqrt{7} + \sqrt{6} \):
\[ \frac{1 \cdot (\sqrt{7} + \sqrt{6})}{(\sqrt{7} - \sqrt{6})(\sqrt{7} + \sqrt{6})} \]

2. **Apply the difference of squares**: The denominator becomes:
\[ \sqrt{7}^2 - \sqrt{6}^2 = 7 - 6 = 1 \]

3. **Simplify**:
\[ \frac{\sqrt{7} + \sqrt{6}}{1} = \sqrt{7} + \sqrt{6} \]

**Final Answer**: \( \sqrt{7} + \sqrt{6} \)

### (iii) Rationalize \( \frac{1}{\sqrt{5} + \sqrt{2}} \)

1. **Multiply by the conjugate**: Multiply the numerator and denominator by \( \sqrt{5} - \sqrt{2} \):
\[ \frac{1 \cdot (\sqrt{5} - \sqrt{2})}{(\sqrt{5} + \sqrt{2})(\sqrt{5} - \sqrt{2})} \]

2. **Apply the difference of squares**: The denominator becomes:
\[ \sqrt{5}^2 - \sqrt{2}^2 = 5 - 2 = 3 \]

3. **Simplify**:
\[ \frac{\sqrt{5} - \sqrt{2}}{3} \]

**Final Answer**: \( \frac{\sqrt{5} - \sqrt{2}}{3} \)

### (iv) Rationalize \( \frac{1}{\sqrt{7} - 2} \)

1. **Multiply by the conjugate**: Multiply the numerator and denominator by \( \sqrt{7} + 2 \):
\[ \frac{1 \cdot (\sqrt{7} + 2)}{(\sqrt{7} - 2)(\sqrt{7} + 2)} \]

2. **Apply the difference of squares**: The denominator becomes:
\[ \sqrt{7}^2 - 2^2 = 7 - 4 = 3 \]

3. **Simplify**:
\[ \frac{\sqrt{7} + 2}{3} \]

**Final Answer**: \( \frac{\sqrt{7} + 2}{3} \)

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