Write the lowest rationalising factor of :
`7-sqrt(7)`

Answer

To find the lowest rationalizing factor of \( 7 - \sqrt{7} \), we will follow these steps:

### Step 1: Identify the expression
We start with the expression \( 7 - \sqrt{7} \).

### Step 2: Write the conjugate
The lowest rationalizing factor can be found by writing the conjugate of the expression. The conjugate of \( a - b \) is \( a + b \). Here, \( a = 7 \) and \( b = \sqrt{7} \). Therefore, the conjugate is:
\[ 7 + \sqrt{7} \]

### Step 3: Multiply the expression by its conjugate
To verify that \( 7 + \sqrt{7} \) is indeed the rationalizing factor, we can multiply the original expression by its conjugate:
\[ (7 - \sqrt{7})(7 + \sqrt{7}) \]

### Step 4: Use the difference of squares identity
This multiplication can be simplified using the difference of squares identity, which states that:
\[ (a - b)(a + b) = a^2 - b^2 \]
Here, \( a = 7 \) and \( b = \sqrt{7} \). Thus, we have:
\[ 7^2 - (\sqrt{7})^2 \]

### Step 5: Calculate the squares
Calculating the squares gives us:
\[ 49 - 7 = 42 \]

### Conclusion
The result of the multiplication is \( 42 \), which is a rational number. Therefore, the lowest rationalizing factor of \( 7 - \sqrt{7} \) is:
\[ \boxed{7 + \sqrt{7}} \]

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