What is the rationalisation factor of `(1)/(3+sqrt(5))` ?

Explanation

To find the rationalization factor of the expression \( \frac{1}{3 + \sqrt{5}} \), we will follow these steps:

### Step 1: Identify the Denominator
The denominator of the expression is \( 3 + \sqrt{5} \).

### Step 2: Find the Conjugate
The conjugate of \( 3 + \sqrt{5} \) is \( 3 - \sqrt{5} \). The conjugate is formed by changing the sign of the square root term.

### Step 3: Multiply and Divide by the Conjugate
To rationalize the denominator, we multiply both the numerator and the denominator by the conjugate:
\[ \frac{1}{3 + \sqrt{5}} \times \frac{3 - \sqrt{5}}{3 - \sqrt{5}} = \frac{3 - \sqrt{5}}{(3 + \sqrt{5})(3 - \sqrt{5})} \]

### Step 4: Simplify the Denominator
Now, we simplify the denominator using the difference of squares formula \( (a + b)(a - b) = a^2 - b^2 \):
\[ (3 + \sqrt{5})(3 - \sqrt{5}) = 3^2 - (\sqrt{5})^2 = 9 - 5 = 4 \]

### Step 5: Write the Rationalized Expression
Now, substituting back into our expression, we have:
\[ \frac{3 - \sqrt{5}}{4} \]

### Step 6: Identify the Rationalization Factor
The rationalization factor is the term we multiplied by to rationalize the denominator, which is \( 3 - \sqrt{5} \).

### Final Answer
Thus, the rationalization factor of \( \frac{1}{3 + \sqrt{5}} \) is:
\[ \boxed{3 - \sqrt{5}} \]

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