Write the lowest rationalising factor of :
`sqrt(5)-3`

Answer

To find the lowest rationalizing factor of the expression \( \sqrt{5} - 3 \), we can follow these steps:

### Step 1: Identify the expression
We have the expression \( \sqrt{5} - 3 \).

### Step 2: Determine the conjugate
The rationalizing factor (or conjugate) of an expression of the form \( a - b \) is \( a + b \). Here, \( a = \sqrt{5} \) and \( b = 3 \). Therefore, the conjugate is:
\[ \sqrt{5} + 3 \]

### Step 3: Multiply the expression by its conjugate
To rationalize \( \sqrt{5} - 3 \), we multiply it by its conjugate:
\[ (\sqrt{5} - 3)(\sqrt{5} + 3) \]

### Step 4: Apply the difference of squares formula
Using the difference of squares formula \( (a - b)(a + b) = a^2 - b^2 \), we can calculate:
\[ (\sqrt{5})^2 - (3)^2 = 5 - 9 \]
This simplifies to:
\[ 5 - 9 = -4 \]

### Step 5: Conclusion
Thus, the product \( (\sqrt{5} - 3)(\sqrt{5} + 3) = -4 \), which is a rational number. Therefore, the lowest rationalizing factor of \( \sqrt{5} - 3 \) is:
\[ \sqrt{5} + 3 \]

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