Rationalise the denominator of `(11)/(5+sqrt(3))`

Explanation

To rationalize the denominator of the expression \( \frac{11}{5 + \sqrt{3}} \), we can follow these steps:

### Step 1: Identify the expression
We have the expression:
\[ \frac{11}{5 + \sqrt{3}} \]

### 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 \( 5 + \sqrt{3} \) is \( 5 - \sqrt{3} \). Thus, we multiply:
\[ \frac{11}{5 + \sqrt{3}} \cdot \frac{5 - \sqrt{3}}{5 - \sqrt{3}} \]

### Step 3: Perform the multiplication
Now, we multiply the numerators and the denominators:
\[ = \frac{11(5 - \sqrt{3})}{(5 + \sqrt{3})(5 - \sqrt{3})} \]

### Step 4: Simplify the denominator
Using the difference of squares formula \( (a + b)(a - b) = a^2 - b^2 \), we calculate the denominator:
\[ = 5^2 - (\sqrt{3})^2 = 25 - 3 = 22 \]

### Step 5: Expand the numerator
Now, we expand the numerator:
\[ = 11(5 - \sqrt{3}) = 55 - 11\sqrt{3} \]

### Step 6: Combine the results
Putting it all together, we have:
\[ = \frac{55 - 11\sqrt{3}}{22} \]

### Step 7: Simplify the fraction
We can simplify this fraction by dividing both terms in the numerator by 22:
\[ = \frac{55}{22} - \frac{11\sqrt{3}}{22} = \frac{5}{2} - \frac{\sqrt{3}}{2} \]

### Final Answer
Thus, the rationalized form of the expression \( \frac{11}{5 + \sqrt{3}} \) is:
\[ \frac{5}{2} - \frac{\sqrt{3}}{2} \]

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