Rationlise the denominator and simplify:
` ( 12sqrt2)/( sqrt3+ sqrt6)`

Answer

To rationalize the denominator and simplify the expression \( \frac{12\sqrt{2}}{\sqrt{3} + \sqrt{6}} \), follow these steps:

### Step 1: Multiply by the Conjugate
To rationalize the denominator, we multiply the numerator and the denominator by the conjugate of the denominator, which is \( \sqrt{3} - \sqrt{6} \).

\[ \frac{12\sqrt{2}}{\sqrt{3} + \sqrt{6}} \times \frac{\sqrt{3} - \sqrt{6}}{\sqrt{3} - \sqrt{6}} = \frac{12\sqrt{2}(\sqrt{3} - \sqrt{6})}{(\sqrt{3} + \sqrt{6})(\sqrt{3} - \sqrt{6})} \]

### Step 2: Simplify the Denominator
Now, we simplify the denominator using the difference of squares formula:

\[ (\sqrt{3})^2 - (\sqrt{6})^2 = 3 - 6 = -3 \]

So, the expression becomes:

\[ \frac{12\sqrt{2}(\sqrt{3} - \sqrt{6})}{-3} \]

### Step 3: Simplify the Numerator
Now, we can simplify the numerator:

\[ 12\sqrt{2}(\sqrt{3} - \sqrt{6}) = 12\sqrt{6} - 12\sqrt{12} \]

### Step 4: Combine and Simplify
Now we can write the expression as:

\[ \frac{12\sqrt{2}(\sqrt{3} - \sqrt{6})}{-3} = -4\sqrt{2}(\sqrt{3} - \sqrt{6}) \]

Distributing the negative sign gives us:

\[ -4\sqrt{6} + 4\sqrt{3} \]

### Final Answer
Thus, the simplified expression is:

\[ 4\sqrt{3} - 4\sqrt{6} \]

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