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 \(\frac{\sqrt{2}+1}{\sqrt{2}-1} + \frac{\sqrt{2}-1}{\sqrt{2}+1}\)

**Step 1:** Rationalize the first term \(\frac{\sqrt{2}+1}{\sqrt{2}-1}\).

To rationalize, multiply the numerator and the denominator by the conjugate of the denominator:

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

**Step 2:** Calculate the numerator and denominator.

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

Denominator:
\[ (\sqrt{2})^2 - (1)^2 = 2 - 1 = 1 \]

So,
\[ \frac{\sqrt{2}+1}{\sqrt{2}-1} = 3 + 2\sqrt{2} \]

**Step 3:** Rationalize the second term \(\frac{\sqrt{2}-1}{\sqrt{2}+1}\).

Multiply the numerator and the denominator by the conjugate of the denominator:

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

**Step 4:** Calculate the numerator and denominator.

Numerator:
\[ (\sqrt{2}-1)^2 = 2 - 2\sqrt{2} + 1 = 3 - 2\sqrt{2} \]

Denominator:
\[ (\sqrt{2})^2 - (1)^2 = 2 - 1 = 1 \]

So,
\[ \frac{\sqrt{2}-1}{\sqrt{2}+1} = 3 - 2\sqrt{2} \]

**Step 5:** Combine both results.

Now, we can add the two simplified terms:
\[ (3 + 2\sqrt{2}) + (3 - 2\sqrt{2}) = 6 \]

**Final Answer for (i):** \(6\)

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### (ii) Simplify \(\frac{\sqrt{5}+\sqrt{3}}{\sqrt{5}-\sqrt{3}} + \frac{\sqrt{5}-\sqrt{3}}{\sqrt{5}+\sqrt{3}}\)

**Step 1:** Rationalize the first term \(\frac{\sqrt{5}+\sqrt{3}}{\sqrt{5}-\sqrt{3}}\).

Multiply the numerator and the denominator by the conjugate of the denominator:

\[ \frac{(\sqrt{5}+\sqrt{3})(\sqrt{5}+\sqrt{3})}{(\sqrt{5}-\sqrt{3})(\sqrt{5}+\sqrt{3})} = \frac{(\sqrt{5}+\sqrt{3})^2}{(\sqrt{5})^2 - (\sqrt{3})^2} \]

**Step 2:** Calculate the numerator and denominator.

Numerator:
\[ (\sqrt{5}+\sqrt{3})^2 = 5 + 2\sqrt{15} + 3 = 8 + 2\sqrt{15} \]

Denominator:
\[ (\sqrt{5})^2 - (\sqrt{3})^2 = 5 - 3 = 2 \]

So,
\[ \frac{\sqrt{5}+\sqrt{3}}{\sqrt{5}-\sqrt{3}} = \frac{8 + 2\sqrt{15}}{2} = 4 + \sqrt{15} \]

**Step 3:** Rationalize the second term \(\frac{\sqrt{5}-\sqrt{3}}{\sqrt{5}+\sqrt{3}}\).

Multiply the numerator and the denominator by the conjugate of the denominator:

\[ \frac{(\sqrt{5}-\sqrt{3})(\sqrt{5}-\sqrt{3})}{(\sqrt{5}+\sqrt{3})(\sqrt{5}-\sqrt{3})} = \frac{(\sqrt{5}-\sqrt{3})^2}{(\sqrt{5})^2 - (\sqrt{3})^2} \]

**Step 4:** Calculate the numerator and denominator.

Numerator:
\[ (\sqrt{5}-\sqrt{3})^2 = 5 - 2\sqrt{15} + 3 = 8 - 2\sqrt{15} \]

Denominator:
\[ (\sqrt{5})^2 - (\sqrt{3})^2 = 5 - 3 = 2 \]

So,
\[ \frac{\sqrt{5}-\sqrt{3}}{\sqrt{5}+\sqrt{3}} = \frac{8 - 2\sqrt{15}}{2} = 4 - \sqrt{15} \]

**Step 5:** Combine both results.

Now, we can add the two simplified terms:
\[ (4 + \sqrt{15}) + (4 - \sqrt{15}) = 8 \]

**Final Answer for (ii):** \(8\)

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### (iii) Simplify \(\frac{2}{\sqrt{5}+\sqrt{3}} + \frac{1}{\sqrt{3}+\sqrt{2}} - \frac{3}{\sqrt{5}+\sqrt{2}}\)

**Step 1:** Rationalize the first term \(\frac{2}{\sqrt{5}+\sqrt{3}}\).

Multiply the numerator and the denominator by the conjugate of the denominator:

\[ \frac{2(\sqrt{5}-\sqrt{3})}{(\sqrt{5}+\sqrt{3})(\sqrt{5}-\sqrt{3})} = \frac{2(\sqrt{5}-\sqrt{3})}{5 - 3} = \frac{2(\sqrt{5}-\sqrt{3})}{2} = \sqrt{5} - \sqrt{3} \]

**Step 2:** Rationalize the second term \(\frac{1}{\sqrt{3}+\sqrt{2}}\).

Multiply the numerator and the denominator by the conjugate of the denominator:

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

**Step 3:** Rationalize the third term \(-\frac{3}{\sqrt{5}+\sqrt{2}}\).

Multiply the numerator and the denominator by the conjugate of the denominator:

\[ -\frac{3(\sqrt{5}-\sqrt{2})}{(\sqrt{5}+\sqrt{2})(\sqrt{5}-\sqrt{2})} = -\frac{3(\sqrt{5}-\sqrt{2})}{5 - 2} = -\frac{3(\sqrt{5}-\sqrt{2})}{3} = -(\sqrt{5}-\sqrt{2}) = -\sqrt{5} + \sqrt{2} \]

**Step 4:** Combine all the results.

Now, we can add:
\[ (\sqrt{5} - \sqrt{3}) + (\sqrt{3} - \sqrt{2}) + (-\sqrt{5} + \sqrt{2}) \]

**Step 5:** Simplify the expression.

\[ \sqrt{5} - \sqrt{5} - \sqrt{3} + \sqrt{3} - \sqrt{2} + \sqrt{2} = 0 \]

**Final Answer for (iii):** \(0\)

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### (iv) Simplify \(\frac{\sqrt{7}+\sqrt{5}}{\sqrt{7}-\sqrt{5}} - \frac{\sqrt{7}-\sqrt{5}}{\sqrt{7}+\sqrt{5}}\)

**Step 1:** Rationalize the first term \(\frac{\sqrt{7}+\sqrt{5}}{\sqrt{7}-\sqrt{5}}\).

Multiply the numerator and the denominator by the conjugate of the denominator:

\[ \frac{(\sqrt{7}+\sqrt{5})(\sqrt{7}+\sqrt{5})}{(\sqrt{7}-\sqrt{5})(\sqrt{7}+\sqrt{5})} = \frac{(\sqrt{7}+\sqrt{5})^2}{(\sqrt{7})^2 - (\sqrt{5})^2} \]

**Step 2:** Calculate the numerator and denominator.

Numerator:
\[ (\sqrt{7}+\sqrt{5})^2 = 7 + 2\sqrt{35} + 5 = 12 + 2\sqrt{35} \]

Denominator:
\[ (\sqrt{7})^2 - (\sqrt{5})^2 = 7 - 5 = 2 \]

So,
\[ \frac{\sqrt{7}+\sqrt{5}}{\sqrt{7}-\sqrt{5}} = \frac{12 + 2\sqrt{35}}{2} = 6 + \sqrt{35} \]

**Step 3:** Rationalize the second term \(\frac{\sqrt{7}-\sqrt{5}}{\sqrt{7}+\sqrt{5}}\).

Multiply the numerator and the denominator by the conjugate of the denominator:

\[ \frac{(\sqrt{7}-\sqrt{5})(\sqrt{7}-\sqrt{5})}{(\sqrt{7}+\sqrt{5})(\sqrt{7}-\sqrt{5})} = \frac{(\sqrt{7}-\sqrt{5})^2}{(\sqrt{7})^2 - (\sqrt{5})^2} \]

**Step 4:** Calculate the numerator and denominator.

Numerator:
\[ (\sqrt{7}-\sqrt{5})^2 = 7 - 2\sqrt{35} + 5 = 12 - 2\sqrt{35} \]

Denominator:
\[ (\sqrt{7})^2 - (\sqrt{5})^2 = 7 - 5 = 2 \]

So,
\[ \frac{\sqrt{7}-\sqrt{5}}{\sqrt{7}+\sqrt{5}} = \frac{12 - 2\sqrt{35}}{2} = 6 - \sqrt{35} \]

**Step 5:** Combine both results.

Now, we can subtract:
\[ (6 + \sqrt{35}) - (6 - \sqrt{35}) = 6 + \sqrt{35} - 6 + \sqrt{35} = 2\sqrt{35} \]

**Final Answer for (iv):** \(2\sqrt{35}\)

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