Rationalise the denominator of the following :
`(1)/(sqrt(3)-sqrt(2)-1)" "(1)/(sqrt(2)+sqrt(3)+sqrt(10))`

Explanation

To rationalize the denominators of the given expressions, we will follow a systematic approach. Let's break down the solution step by step.

### Part 1: Rationalizing \( \frac{1}{\sqrt{3} - \sqrt{2} - 1} \)

1. **Identify the Denominator**: The denominator is \( \sqrt{3} - \sqrt{2} - 1 \).

2. **Multiply by the Conjugate**: We will multiply the numerator and denominator by the conjugate of the first two terms, which is \( \sqrt{3} - \sqrt{2} + 1 \).
\[ \frac{1}{\sqrt{3} - \sqrt{2} - 1} \cdot \frac{\sqrt{3} - \sqrt{2} + 1}{\sqrt{3} - \sqrt{2} + 1} \]

3. **Simplify the Denominator**: The denominator becomes:
\[ (\sqrt{3} - \sqrt{2})^2 - 1^2 \]
Expanding this:
\[ = (\sqrt{3})^2 - 2\sqrt{3}\sqrt{2} + (\sqrt{2})^2 - 1 \]
\[ = 3 - 2\sqrt{6} + 2 - 1 = 4 - 2\sqrt{6} \]

4. **Simplify the Numerator**: The numerator is:
\[ \sqrt{3} - \sqrt{2} + 1 \]

5. **Final Expression**: Thus, we have:
\[ \frac{\sqrt{3} - \sqrt{2} + 1}{4 - 2\sqrt{6}} \]

6. **Rationalize Again**: Now, we will rationalize \( \frac{\sqrt{3} - \sqrt{2} + 1}{4 - 2\sqrt{6}} \) by multiplying by the conjugate of the denominator:
\[ \frac{\sqrt{3} - \sqrt{2} + 1}{4 - 2\sqrt{6}} \cdot \frac{4 + 2\sqrt{6}}{4 + 2\sqrt{6}} \]

7. **Denominator Calculation**: The denominator becomes:
\[ (4)^2 - (2\sqrt{6})^2 = 16 - 24 = -8 \]

8. **Numerator Calculation**: The numerator expands to:
\[ (4)(\sqrt{3} - \sqrt{2} + 1) + (2\sqrt{6})(\sqrt{3} - \sqrt{2} + 1) \]
Simplifying:
\[ 4\sqrt{3} - 4\sqrt{2} + 4 + 2\sqrt{18} - 2\sqrt{12} + 2\sqrt{6} \]
\[ = 4\sqrt{3} - 4\sqrt{2} + 4 + 6\sqrt{2} - 4\sqrt{3} + 2\sqrt{6} \]
\[ = (4 - 4)\sqrt{3} + (6 - 4)\sqrt{2} + 4 + 2\sqrt{6} = 2\sqrt{2} + 4 + 2\sqrt{6} \]

9. **Final Result**: Therefore, the expression simplifies to:
\[ \frac{2\sqrt{2} + 4 + 2\sqrt{6}}{-8} = -\frac{1}{4}(\sqrt{2} + 2 + \sqrt{6}) \]

### Part 2: Rationalizing \( \frac{1}{\sqrt{2} + \sqrt{3} + \sqrt{10}} \)

1. **Identify the Denominator**: The denominator is \( \sqrt{2} + \sqrt{3} + \sqrt{10} \).

2. **Multiply by the Conjugate**: We will multiply the numerator and denominator by the conjugate \( \sqrt{2} + \sqrt{3} - \sqrt{10} \):
\[ \frac{1}{\sqrt{2} + \sqrt{3} + \sqrt{10}} \cdot \frac{\sqrt{2} + \sqrt{3} - \sqrt{10}}{\sqrt{2} + \sqrt{3} - \sqrt{10}} \]

3. **Simplify the Denominator**: The denominator becomes:
\[ (\sqrt{2} + \sqrt{3})^2 - (\sqrt{10})^2 \]
Expanding this:
\[ = (2 + 3 + 2\sqrt{6}) - 10 = -5 + 2\sqrt{6} \]

4. **Simplify the Numerator**: The numerator is:
\[ \sqrt{2} + \sqrt{3} - \sqrt{10} \]

5. **Final Expression**: Thus, we have:
\[ \frac{\sqrt{2} + \sqrt{3} - \sqrt{10}}{-5 + 2\sqrt{6}} \]

6. **Rationalize Again**: Now, we will rationalize \( \frac{\sqrt{2} + \sqrt{3} - \sqrt{10}}{-5 + 2\sqrt{6}} \) by multiplying by the conjugate of the denominator:
\[ \frac{\sqrt{2} + \sqrt{3} - \sqrt{10}}{-5 + 2\sqrt{6}} \cdot \frac{-5 - 2\sqrt{6}}{-5 - 2\sqrt{6}} \]

7. **Denominator Calculation**: The denominator becomes:
\[ (-5)^2 - (2\sqrt{6})^2 = 25 - 24 = 1 \]

8. **Numerator Calculation**: The numerator expands to:
\[ (-5)(\sqrt{2} + \sqrt{3} - \sqrt{10}) - (2\sqrt{6})(\sqrt{2} + \sqrt{3} - \sqrt{10}) \]
Simplifying:
\[ -5\sqrt{2} - 5\sqrt{3} + 5\sqrt{10} - 2\sqrt{12} - 2\sqrt{18} + 2\sqrt{60} \]
\[ = -5\sqrt{2} - 5\sqrt{3} + 5\sqrt{10} - 4\sqrt{3} - 6\sqrt{2} + 2\sqrt{15} \]
\[ = (-5 - 6)\sqrt{2} + (-5 - 4)\sqrt{3} + 5\sqrt{10} + 2\sqrt{15} \]
\[ = -11\sqrt{2} - 9\sqrt{3} + 5\sqrt{10} + 2\sqrt{15} \]

9. **Final Result**: Therefore, the expression simplifies to:
\[ -11\sqrt{2} - 9\sqrt{3} + 5\sqrt{10} + 2\sqrt{15} \]