Simplify each of the following :
`(i)(sqrt(2)+1)/(sqrt(2)-1)+(sqrt(2)-1)/(sqrt(2)+1)" "(ii)(sqrt(5)+sqrt(3))/(sqrt(5)-sqrt(3))+(sqrt(5)-sqrt(3))/(sqrt(5)+sqrt(3))" "(iii)(2)/(sqrt(5)+sqrt(3))+(1)/(sqrt(3)+sqrt(2))-(3)/(sqrt(5)+sqrt(2))" "(iv)(sqrt(7)+sqrt(5))/(sqrt(7)-sqrt(5))-(sqrt(7)-sqrt(5))/(sqrt(7)+sqrt(5))`

Explanation

Let's simplify each of the given expressions step by step.

### (i) Simplify \(\frac{\sqrt{2}+1}{\sqrt{2}-1} + \frac{\sqrt{2}-1}{\sqrt{2}+1}\)

1. **Find a common denominator**:
The common denominator for the two fractions is \((\sqrt{2}-1)(\sqrt{2}+1)\).

2. **Rewrite the fractions**:
\[ \frac{(\sqrt{2}+1)(\sqrt{2}+1) + (\sqrt{2}-1)(\sqrt{2}-1)}{(\sqrt{2}-1)(\sqrt{2}+1)} \]

3. **Expand the numerators**:
\[ (\sqrt{2}+1)^2 = 2 + 1 + 2\sqrt{2} = 3 + 2\sqrt{2} \]
\[ (\sqrt{2}-1)^2 = 2 + 1 - 2\sqrt{2} = 3 - 2\sqrt{2} \]

4. **Combine the numerators**:
\[ (3 + 2\sqrt{2}) + (3 - 2\sqrt{2}) = 6 \]

5. **Simplify the denominator**:
\[ (\sqrt{2}-1)(\sqrt{2}+1) = 2 - 1 = 1 \]

6. **Final result**:
\[ \frac{6}{1} = 6 \]