Rationalize the denominatior of :
`(sqrt(3)+1)/(sqrt(3)-1)`

Answer

To rationalize the denominator of the expression \((\sqrt{3} + 1) / (\sqrt{3} - 1)\), we will follow these steps:

### Step 1: Identify the expression
We have the expression:
\[ \frac{\sqrt{3} + 1}{\sqrt{3} - 1} \]

### Step 2: Multiply numerator and denominator by the conjugate of the denominator
To rationalize the denominator, we multiply both the numerator and the denominator by the conjugate of the denominator, which is \((\sqrt{3} + 1)\):
\[ \frac{(\sqrt{3} + 1)(\sqrt{3} + 1)}{(\sqrt{3} - 1)(\sqrt{3} + 1)} \]

### Step 3: Simplify the numerator
Using the identity \((a + b)^2 = a^2 + 2ab + b^2\), we can simplify the numerator:
\[ (\sqrt{3} + 1)^2 = (\sqrt{3})^2 + 2(\sqrt{3})(1) + (1)^2 = 3 + 2\sqrt{3} + 1 = 4 + 2\sqrt{3} \]

### Step 4: Simplify the denominator
Using the identity \(a^2 - b^2 = (a - b)(a + b)\), we simplify the denominator:
\[ (\sqrt{3})^2 - (1)^2 = 3 - 1 = 2 \]

### Step 5: Combine the results
Now we can combine the simplified numerator and denominator:
\[ \frac{4 + 2\sqrt{3}}{2} \]

### Step 6: Simplify the fraction
We can simplify this fraction by dividing both terms in the numerator by 2:
\[ \frac{4}{2} + \frac{2\sqrt{3}}{2} = 2 + \sqrt{3} \]

### Final Answer
Thus, the rationalized form of the expression is:
\[ 2 + \sqrt{3} \]

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