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

Answer

To rationalize the denominator and simplify the expression \((\sqrt{5} + \sqrt{3}) / (\sqrt{5} - \sqrt{3})\), follow these steps:

### Step 1: Multiply by the Conjugate
Multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of \((\sqrt{5} - \sqrt{3})\) is \((\sqrt{5} + \sqrt{3})\).

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

### Step 2: Simplify the Denominator
Use the difference of squares formula \(a^2 - b^2\) to simplify the denominator:

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

### Step 3: Expand the Numerator
Now expand the numerator \((\sqrt{5} + \sqrt{3})^2\):

\[ (\sqrt{5} + \sqrt{3})^2 = (\sqrt{5})^2 + 2(\sqrt{5})(\sqrt{3}) + (\sqrt{3})^2 = 5 + 2\sqrt{15} + 3 = 8 + 2\sqrt{15} \]

### Step 4: Combine the Results
Now, substitute the simplified numerator and denominator back into the expression:

\[ \frac{8 + 2\sqrt{15}}{2} \]

### Step 5: Simplify the Expression
Divide each term in the numerator by the denominator:

\[ \frac{8}{2} + \frac{2\sqrt{15}}{2} = 4 + \sqrt{15} \]

### Final Result
Thus, the simplified expression is:

\[ \frac{\sqrt{5} + \sqrt{3}}{\sqrt{5} - \sqrt{3}} = 4 + \sqrt{15} \]

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