Write the lowest rationalising factor of :
`3 sqrt(2)+ 2sqrt(3)`

Answer

To find the lowest rationalizing factor of the expression \(3\sqrt{2} + 2\sqrt{3}\), we will follow these steps:

### Step 1: Identify the expression
We have the expression \(3\sqrt{2} + 2\sqrt{3}\).

### Step 2: Write the conjugate
The lowest rationalizing factor is obtained by writing the conjugate of the expression. The conjugate of \(a + b\) is \(a - b\). Therefore, the conjugate of \(3\sqrt{2} + 2\sqrt{3}\) is \(3\sqrt{2} - 2\sqrt{3}\).

### Step 3: Multiply the expression by its conjugate
To rationalize the expression, we multiply \(3\sqrt{2} + 2\sqrt{3}\) by its conjugate \(3\sqrt{2} - 2\sqrt{3}\):

\[ (3\sqrt{2} + 2\sqrt{3})(3\sqrt{2} - 2\sqrt{3}) \]

### Step 4: Apply the difference of squares formula
Using the identity \( (a + b)(a - b) = a^2 - b^2 \), we can simplify the multiplication:

Let \(a = 3\sqrt{2}\) and \(b = 2\sqrt{3}\). Then, we have:

\[ (3\sqrt{2})^2 - (2\sqrt{3})^2 \]

### Step 5: Calculate \(a^2\) and \(b^2\)
Now, we calculate \(a^2\) and \(b^2\):

1. \( (3\sqrt{2})^2 = 9 \cdot 2 = 18 \)
2. \( (2\sqrt{3})^2 = 4 \cdot 3 = 12 \)

### Step 6: Subtract the results
Now we subtract the two results:

\[ 18 - 12 = 6 \]

### Conclusion
Thus, the lowest rationalizing factor of \(3\sqrt{2} + 2\sqrt{3}\) is \(3\sqrt{2} - 2\sqrt{3}\).

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