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 problems, we will simplify each expression step by step and compare the results to find the values of \( a \) and \( b \).

### Part (i):
We need to find \( a \) and \( b \) such that:

\[ \frac{\sqrt{3}+1}{\sqrt{3}-1} = a + b\sqrt{3} \]

**Step 1: Rationalize the denominator.**

To simplify the left-hand side, we can multiply the numerator and the denominator by the conjugate of the denominator:

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

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

The numerator becomes:

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

The denominator becomes:

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

So, we have:

\[ \frac{4 + 2\sqrt{3}}{2} \]

**Step 3: Simplify the expression.**

Dividing both terms in the numerator by 2 gives:

\[ 2 + \sqrt{3} \]

**Step 4: Compare with \( a + b\sqrt{3} \).**

Now we can compare:

\[ 2 + \sqrt{3} = a + b\sqrt{3} \]

From this, we can see that:

- \( a = 2 \)
- \( b = 1 \)

### Part (ii):
Now we need to find \( a \) and \( b \) for the expression:

\[ \frac{5 + 2\sqrt{3}}{5 - 2\sqrt{3}} = a + b\sqrt{3} \]

**Step 1: Rationalize the denominator.**

Again, we 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: Expand the numerator and denominator.**

The numerator becomes:

\[ (5 + 2\sqrt{3})(5 + 2\sqrt{3}) = 5^2 + 2 \cdot 5 \cdot 2\sqrt{3} + (2\sqrt{3})^2 = 25 + 20\sqrt{3} + 12 = 37 + 20\sqrt{3} \]

The denominator becomes:

\[ (5 - 2\sqrt{3})(5 + 2\sqrt{3}) = 5^2 - (2\sqrt{3})^2 = 25 - 12 = 13 \]

So, we have:

\[ \frac{37 + 20\sqrt{3}}{13} \]

**Step 3: Simplify the expression.**

Dividing both terms in the numerator by 13 gives:

\[ \frac{37}{13} + \frac{20\sqrt{3}}{13} \]

**Step 4: Compare with \( a + b\sqrt{3} \).**

Now we can compare:

\[ \frac{37}{13} + \frac{20\sqrt{3}}{13} = a + b\sqrt{3} \]

From this, we can see that:

- \( a = \frac{37}{13} \)
- \( b = \frac{20}{13} \)

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