Rationalise the denominator : `(5)/(sqrt(11)+4)`.

Explanation

To rationalize the denominator of the expression \(\frac{5}{\sqrt{11}+4}\), we will follow these steps:

### Step 1: Identify the expression
We start with the expression:
\[ \frac{5}{\sqrt{11}+4} \]

### 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 \(\sqrt{11}+4\) is \(\sqrt{11}-4\). Thus, we multiply:
\[ \frac{5}{\sqrt{11}+4} \cdot \frac{\sqrt{11}-4}{\sqrt{11}-4} \]

### Step 3: Simplify the numerator
Now, we simplify the numerator:
\[ 5(\sqrt{11}-4) = 5\sqrt{11} - 20 \]

### Step 4: Simplify the denominator
Next, we simplify the denominator using the difference of squares formula:
\[ (\sqrt{11}+4)(\sqrt{11}-4) = (\sqrt{11})^2 - (4)^2 = 11 - 16 = -5 \]

### Step 5: Combine the results
Now we can combine the results:
\[ \frac{5\sqrt{11} - 20}{-5} \]

### Step 6: Simplify the fraction
We can simplify this expression by dividing both terms in the numerator by \(-5\):
\[ \frac{5\sqrt{11}}{-5} - \frac{20}{-5} = -\sqrt{11} + 4 \]
This can be rewritten as:
\[ 4 - \sqrt{11} \]

### Final Answer
Thus, the rationalized form of the expression \(\frac{5}{\sqrt{11}+4}\) is:
\[ 4 - \sqrt{11} \]

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