Rationalise the denominator of each the of the following :
`(i)(1)/(3+sqrt(5))" "(ii)(1)/(sqrt(5)-sqrt(3))" "(iii)(16)/(sqrt(41)+5)" "(iv)(30)/(5sqrt(3)+3sqrt(5))" "(v)(3-2sqrt(2))/(3+2sqrt(2))`

Explanation

Let's rationalize the denominators of the given expressions step by step.

### (i) Rationalizing \( \frac{1}{3 + \sqrt{5}} \)

**Step 1:** Multiply the numerator and denominator by the conjugate of the denominator, which is \( 3 - \sqrt{5} \).

\[ \frac{1}{3 + \sqrt{5}} \cdot \frac{3 - \sqrt{5}}{3 - \sqrt{5}} = \frac{3 - \sqrt{5}}{(3 + \sqrt{5})(3 - \sqrt{5})} \]

**Step 2:** Simplify the denominator using the difference of squares formula \( (a + b)(a - b) = a^2 - b^2 \).

\[ (3 + \sqrt{5})(3 - \sqrt{5}) = 3^2 - (\sqrt{5})^2 = 9 - 5 = 4 \]

**Step 3:** Substitute back into the expression.

\[ \frac{3 - \sqrt{5}}{4} \]

### (ii) Rationalizing \( \frac{1}{\sqrt{5} - \sqrt{3}} \)

**Step 1:** Multiply by the conjugate \( \sqrt{5} + \sqrt{3} \).

\[ \frac{1}{\sqrt{5} - \sqrt{3}} \cdot \frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} + \sqrt{3}} = \frac{\sqrt{5} + \sqrt{3}}{(\sqrt{5})^2 - (\sqrt{3})^2} \]

**Step 2:** Simplify the denominator.

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

**Step 3:** Substitute back into the expression.

\[ \frac{\sqrt{5} + \sqrt{3}}{2} \]

### (iii) Rationalizing \( \frac{16}{\sqrt{41} + 5} \)

**Step 1:** Multiply by the conjugate \( \sqrt{41} - 5 \).

\[ \frac{16}{\sqrt{41} + 5} \cdot \frac{\sqrt{41} - 5}{\sqrt{41} - 5} = \frac{16(\sqrt{41} - 5)}{(\sqrt{41})^2 - 5^2} \]

**Step 2:** Simplify the denominator.

\[ (\sqrt{41})^2 - 5^2 = 41 - 25 = 16 \]

**Step 3:** Substitute back into the expression.

\[ \frac{16(\sqrt{41} - 5)}{16} = \sqrt{41} - 5 \]

### (iv) Rationalizing \( \frac{30}{5\sqrt{3} + 3\sqrt{5}} \)

**Step 1:** Multiply by the conjugate \( 5\sqrt{3} - 3\sqrt{5} \).

\[ \frac{30}{5\sqrt{3} + 3\sqrt{5}} \cdot \frac{5\sqrt{3} - 3\sqrt{5}}{5\sqrt{3} - 3\sqrt{5}} = \frac{30(5\sqrt{3} - 3\sqrt{5})}{(5\sqrt{3})^2 - (3\sqrt{5})^2} \]

**Step 2:** Simplify the denominator.

\[ (5\sqrt{3})^2 - (3\sqrt{5})^2 = 25 \cdot 3 - 9 \cdot 5 = 75 - 45 = 30 \]

**Step 3:** Substitute back into the expression.

\[ \frac{30(5\sqrt{3} - 3\sqrt{5})}{30} = 5\sqrt{3} - 3\sqrt{5} \]

### (v) Rationalizing \( \frac{3 - 2\sqrt{2}}{3 + 2\sqrt{2}} \)

**Step 1:** Multiply by the conjugate \( 3 - 2\sqrt{2} \).

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

**Step 2:** Simplify the denominator.

\[ (3 + 2\sqrt{2})(3 - 2\sqrt{2}) = 3^2 - (2\sqrt{2})^2 = 9 - 8 = 1 \]

**Step 3:** Substitute back into the expression.

\[ (3 - 2\sqrt{2})^2 = 9 - 12\sqrt{2} + 8 = 17 - 12\sqrt{2} \]

### Final Answers:
1. \( \frac{3 - \sqrt{5}}{4} \)
2. \( \frac{\sqrt{5} + \sqrt{3}}{2} \)
3. \( \sqrt{41} - 5 \)
4. \( 5\sqrt{3} - 3\sqrt{5} \)
5. \( 17 - 12\sqrt{2} \)