Find the value of a and b if
`(i)(sqrt(3)+1)/(sqrt(3)-1)=a+bsqrt(3)" "(ii)(5+2sqrt(3))/(5-2sqrt(3))=a+bsqrt(3)`

Explanation

To solve the given equations step by step, we will find the values of \( a \) and \( b \) for both parts of the question.

### Part (i):
We need to find \( a \) and \( b \) such that:
\[ \frac{\sqrt{3} + 1}{\sqrt{3} - 1} = a + b\sqrt{3} \]

**Step 1:** Multiply the numerator and denominator by the conjugate of the denominator:
\[ \frac{(\sqrt{3} + 1)(\sqrt{3} + 1)}{(\sqrt{3} - 1)(\sqrt{3} + 1)} \]

**Step 2:** Simplify the numerator:
\[ (\sqrt{3} + 1)^2 = 3 + 2\sqrt{3} + 1 = 4 + 2\sqrt{3} \]

**Step 3:** Simplify the denominator:
\[ (\sqrt{3})^2 - (1)^2 = 3 - 1 = 2 \]

**Step 4:** Combine the results:
\[ \frac{4 + 2\sqrt{3}}{2} = \frac{4}{2} + \frac{2\sqrt{3}}{2} = 2 + \sqrt{3} \]

**Step 5:** Compare with \( a + b\sqrt{3} \):
\[ a + b\sqrt{3} = 2 + 1\sqrt{3} \]
Thus, \( a = 2 \) and \( b = 1 \).

### Part (ii):
Now we will solve the second part:
\[ \frac{5 + 2\sqrt{3}}{5 - 2\sqrt{3}} = a + b\sqrt{3} \]

**Step 1:** Multiply the numerator and denominator by the conjugate of the denominator:
\[ \frac{(5 + 2\sqrt{3})(5 + 2\sqrt{3})}{(5 - 2\sqrt{3})(5 + 2\sqrt{3})} \]

**Step 2:** Simplify the numerator:
\[ (5 + 2\sqrt{3})^2 = 25 + 20\sqrt{3} + 12 = 37 + 20\sqrt{3} \]

**Step 3:** Simplify the denominator:
\[ (5)^2 - (2\sqrt{3})^2 = 25 - 12 = 13 \]

**Step 4:** Combine the results:
\[ \frac{37 + 20\sqrt{3}}{13} = \frac{37}{13} + \frac{20\sqrt{3}}{13} \]

**Step 5:** Compare with \( a + b\sqrt{3} \):
\[ a + b\sqrt{3} = \frac{37}{13} + \frac{20}{13}\sqrt{3} \]
Thus, \( a = \frac{37}{13} \) and \( b = \frac{20}{13} \).

### Final Answers:
- For part (i): \( a = 2 \), \( b = 1 \)
- For part (ii): \( a = \frac{37}{13} \), \( b = \frac{20}{13} \)