If `sqrt(2)=1.414,sqrt(3)=1.732`, find the value of the following :
`(i)(sqrt(2)+1)/(sqrt(2)-1)" "(ii)(sqrt(3)-1)/(sqrt(3)+1)" "(iii)(2+sqrt(6))/(sqrt(2))`

Explanation

Let's solve the given problems step by step.

### Given:
- \( \sqrt{2} = 1.414 \)
- \( \sqrt{3} = 1.732 \)

### (i) Find the value of \( \frac{\sqrt{2} + 1}{\sqrt{2} - 1} \)

**Step 1:** Multiply the numerator and denominator by \( \sqrt{2} + 1 \) to rationalize the denominator.

\[ \frac{\sqrt{2} + 1}{\sqrt{2} - 1} \cdot \frac{\sqrt{2} + 1}{\sqrt{2} + 1} = \frac{(\sqrt{2} + 1)^2}{(\sqrt{2})^2 - (1)^2} \]

**Step 2:** Simplify the denominator using the difference of squares formula.

\[ = \frac{(\sqrt{2} + 1)^2}{2 - 1} = \frac{(\sqrt{2} + 1)^2}{1} \]

**Step 3:** Expand the numerator.

\[ (\sqrt{2} + 1)^2 = (\sqrt{2})^2 + 2 \cdot \sqrt{2} \cdot 1 + (1)^2 = 2 + 2\sqrt{2} + 1 = 3 + 2\sqrt{2} \]

**Step 4:** Therefore, the expression simplifies to:

\[ 3 + 2\sqrt{2} \]

**Step 5:** Substitute \( \sqrt{2} \) with \( 1.414 \).

\[ 3 + 2 \cdot 1.414 = 3 + 2.828 = 5.828 \]

### Answer for (i): \( 5.828 \)

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### (ii) Find the value of \( \frac{\sqrt{3} - 1}{\sqrt{3} + 1} \)

**Step 1:** Multiply the numerator and denominator by \( \sqrt{3} - 1 \).

\[ \frac{\sqrt{3} - 1}{\sqrt{3} + 1} \cdot \frac{\sqrt{3} - 1}{\sqrt{3} - 1} = \frac{(\sqrt{3} - 1)^2}{(\sqrt{3})^2 - (1)^2} \]

**Step 2:** Simplify the denominator.

\[ = \frac{(\sqrt{3} - 1)^2}{3 - 1} = \frac{(\sqrt{3} - 1)^2}{2} \]

**Step 3:** Expand the numerator.

\[ (\sqrt{3} - 1)^2 = (\sqrt{3})^2 - 2 \cdot \sqrt{3} \cdot 1 + (1)^2 = 3 - 2\sqrt{3} + 1 = 4 - 2\sqrt{3} \]

**Step 4:** Therefore, the expression simplifies to:

\[ \frac{4 - 2\sqrt{3}}{2} = 2 - \sqrt{3} \]

**Step 5:** Substitute \( \sqrt{3} \) with \( 1.732 \).

\[ 2 - 1.732 = 0.268 \]

### Answer for (ii): \( 0.268 \)

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### (iii) Find the value of \( \frac{2 + \sqrt{6}}{\sqrt{2}} \)

**Step 1:** Multiply the numerator and denominator by \( \sqrt{2} \).

\[ \frac{2 + \sqrt{6}}{\sqrt{2}} \cdot \frac{\sqrt{2}}{\sqrt{2}} = \frac{(2 + \sqrt{6})\sqrt{2}}{2} \]

**Step 2:** Distribute \( \sqrt{2} \) in the numerator.

\[ = \frac{2\sqrt{2} + \sqrt{6} \cdot \sqrt{2}}{2} = \frac{2\sqrt{2} + \sqrt{12}}{2} \]

**Step 3:** Simplify \( \sqrt{12} \).

\[ \sqrt{12} = \sqrt{4 \cdot 3} = 2\sqrt{3} \]

**Step 4:** Substitute back into the expression.

\[ = \frac{2\sqrt{2} + 2\sqrt{3}}{2} = \sqrt{2} + \sqrt{3} \]

**Step 5:** Substitute \( \sqrt{2} \) and \( \sqrt{3} \) with their values.

\[ = 1.414 + 1.732 = 3.146 \]

### Answer for (iii): \( 3.146 \)

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### Summary of Answers:
1. (i) \( 5.828 \)
2. (ii) \( 0.268 \)
3. (iii) \( 3.146 \)

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