Rationlise the denominator and simplify:
` (1)/(2sqrt5-sqrt3) `

Answer

To rationalize the denominator of the expression \( \frac{1}{2\sqrt{5} - \sqrt{3}} \) and simplify it, follow these steps:

### Step 1: Identify the expression
We start with the expression:
\[ \frac{1}{2\sqrt{5} - \sqrt{3}} \]

### Step 2: Multiply by the conjugate
To rationalize the denominator, we multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of \( 2\sqrt{5} - \sqrt{3} \) is \( 2\sqrt{5} + \sqrt{3} \). Thus, we have:
\[ \frac{1}{2\sqrt{5} - \sqrt{3}} \cdot \frac{2\sqrt{5} + \sqrt{3}}{2\sqrt{5} + \sqrt{3}} \]

### Step 3: Simplify the numerator
The numerator becomes:
\[ 1 \cdot (2\sqrt{5} + \sqrt{3}) = 2\sqrt{5} + \sqrt{3} \]

### Step 4: Simplify the denominator using the difference of squares
The denominator is:
\[ (2\sqrt{5} - \sqrt{3})(2\sqrt{5} + \sqrt{3}) = (2\sqrt{5})^2 - (\sqrt{3})^2 \]
Calculating this gives:
\[ (2\sqrt{5})^2 = 4 \cdot 5 = 20 \]
\[ (\sqrt{3})^2 = 3 \]
So, the denominator simplifies to:
\[ 20 - 3 = 17 \]

### Step 5: Combine the results
Now we can write the expression as:
\[ \frac{2\sqrt{5} + \sqrt{3}}{17} \]

### Final Result
Thus, the rationalized and simplified expression is:
\[ \frac{2\sqrt{5} + \sqrt{3}}{17} \]

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