(i) If `x = (6ab)/(a + b)`, find the value of :
`(x + 3a)/(x - 3a) + (x + 3b)/(x - 3b)`.
(ii) `a = (4sqrt(6))/(sqrt(2) + sqrt(3))`, find the value of :
`(a + 2sqrt(2))/(a - 2sqrt(2)) + (a + 2sqrt(3))/(a - 2sqrt(3))`.

Answer

To solve the given problems step by step, let's break down each part of the question.

### Part (i)
Given:
\[ x = \frac{6ab}{a + b} \]
We need to find the value of:
\[ \frac{x + 3a}{x - 3a} + \frac{x + 3b}{x - 3b} \]

**Step 1: Express \( \frac{x + 3a}{x - 3a} \) in terms of \( a \) and \( b \)**

From the given \( x \):
\[ \frac{x}{3a} = \frac{2b}{a + b} \]

Using the property of componendo and dividendo:
\[ \frac{x + 3a}{x - 3a} = \frac{\frac{x}{3a} + 1}{\frac{x}{3a} - 1} = \frac{\frac{2b}{a + b} + 1}{\frac{2b}{a + b} - 1} \]

This simplifies to:
\[ \frac{2b + (a + b)}{2b - (a + b)} = \frac{a + 3b}{b - a} \]

**Step 2: Express \( \frac{x + 3b}{x - 3b} \) similarly**

From the same \( x \):
\[ \frac{x}{3b} = \frac{2a}{a + b} \]

Using the same property:
\[ \frac{x + 3b}{x - 3b} = \frac{\frac{x}{3b} + 1}{\frac{x}{3b} - 1} = \frac{\frac{2a}{a + b} + 1}{\frac{2a}{a + b} - 1} \]

This simplifies to:
\[ \frac{2a + (a + b)}{2a - (a + b)} = \frac{3a + b}{a - b} \]

**Step 3: Add the two fractions**

Now we add the two results:
\[ \frac{a + 3b}{b - a} + \frac{3a + b}{a - b} \]

Notice that \( a - b = - (b - a) \):
\[ = \frac{a + 3b}{b - a} - \frac{3a + b}{b - a} \]
\[ = \frac{(a + 3b) - (3a + b)}{b - a} \]
\[ = \frac{-2a + 2b}{b - a} \]
\[ = \frac{2(b - a)}{b - a} = 2 \]

Thus, the final answer for part (i) is:
\[ \boxed{2} \]

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### Part (ii)
Given:
\[ a = \frac{4\sqrt{6}}{\sqrt{2} + \sqrt{3}} \]
We need to find the value of:
\[ \frac{a + 2\sqrt{2}}{a - 2\sqrt{2}} + \frac{a + 2\sqrt{3}}{a - 2\sqrt{3}} \]

**Step 1: Rationalize \( a \)**

To rationalize \( a \):
\[ a = \frac{4\sqrt{6}(\sqrt{3} - \sqrt{2})}{(\sqrt{2} + \sqrt{3})(\sqrt{3} - \sqrt{2})} \]
The denominator becomes:
\[ 3 - 2 = 1 \]
Thus,
\[ a = 4\sqrt{6}(\sqrt{3} - \sqrt{2}) = 4\sqrt{18} - 4\sqrt{12} = 12\sqrt{2} - 8\sqrt{3} \]

**Step 2: Calculate \( a + 2\sqrt{2} \) and \( a - 2\sqrt{2} \)**

\[ a + 2\sqrt{2} = (12\sqrt{2} - 8\sqrt{3}) + 2\sqrt{2} = 14\sqrt{2} - 8\sqrt{3} \]
\[ a - 2\sqrt{2} = (12\sqrt{2} - 8\sqrt{3}) - 2\sqrt{2} = 10\sqrt{2} - 8\sqrt{3} \]

**Step 3: Calculate \( a + 2\sqrt{3} \) and \( a - 2\sqrt{3} \)**

\[ a + 2\sqrt{3} = (12\sqrt{2} - 8\sqrt{3}) + 2\sqrt{3} = 12\sqrt{2} - 6\sqrt{3} \]
\[ a - 2\sqrt{3} = (12\sqrt{2} - 8\sqrt{3}) - 2\sqrt{3} = 12\sqrt{2} - 10\sqrt{3} \]

**Step 4: Form the fractions**

Now we form the fractions:
\[ \frac{14\sqrt{2} - 8\sqrt{3}}{10\sqrt{2} - 8\sqrt{3}} + \frac{12\sqrt{2} - 6\sqrt{3}}{12\sqrt{2} - 10\sqrt{3}} \]

**Step 5: Simplify each fraction**

Factor out common terms and simplify:
1. For the first fraction, factor out 2:
\[ = \frac{2(7\sqrt{2} - 4\sqrt{3})}{2(5\sqrt{2} - 4\sqrt{3})} = \frac{7\sqrt{2} - 4\sqrt{3}}{5\sqrt{2} - 4\sqrt{3}} \]

2. For the second fraction:
\[ = \frac{12\sqrt{2} - 6\sqrt{3}}{12\sqrt{2} - 10\sqrt{3}} \]

**Step 6: Add the two fractions**

Combine the fractions:
\[ = \frac{(7\sqrt{2} - 4\sqrt{3})(12\sqrt{2} - 10\sqrt{3}) + (12\sqrt{2} - 6\sqrt{3})(5\sqrt{2} - 4\sqrt{3})}{(5\sqrt{2} - 4\sqrt{3})(12\sqrt{2} - 10\sqrt{3})} \]

After simplification, we find that the sum equals 2.

Thus, the final answer for part (ii) is:
\[ \boxed{2} \]

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