-2/2 root 3 + 6 . take note that -2 is the numerator and 2 root 3 plus 6 is the denominator, it is dealing with conjugate of surds

Let's simplify the given expression by rationalizing the denominator. The expression is:

\[
\frac{-2}{2\sqrt{3} + 6}
\]

To rationalize the denominator, we multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of \(2\sqrt{3} + 6\) is \(2\sqrt{3} - 6\). Therefore, we have:

\[
\frac{-2}{2\sqrt{3} + 6} \times \frac{2\sqrt{3} - 6}{2\sqrt{3} - 6}
\]

Now, let's simplify this expression step by step.

### Step 1: Multiply the Numerator
\[
\text{Numerator: } -2 \times (2\sqrt{3} - 6) = -4\sqrt{3} + 12
\]

### Step 2: Multiply the Denominator
\[
\text{Denominator: } (2\sqrt{3} + 6)(2\sqrt{3} - 6)
\]

This is a difference of squares, so:

\[
(2\sqrt{3})^2 - 6^2 = 4 \times 3 - 36 = 12 - 36 = -24
\]

### Step 3: Combine and Simplify
\[
\frac{-4\sqrt{3} + 12}{-24}
\]

This simplifies by dividing both the numerator and denominator by \(-4\):

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

So, the simplified expression is:

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

Would you like further details or explanations? 

Here are 5 related questions:
1. How do you identify the conjugate of a surd expression?
2. What is the significance of rationalizing the denominator in mathematical expressions?
3. Can you explain the difference of squares and its applications?
4. How would you handle a more complex surd in the denominator?
5. What are some common mistakes to avoid when working with surds?

**Tip:** When simplifying expressions involving surds, always check if further simplification is possible by factoring out common terms in the numerator and denominator.

Learn how to simplify the expression -2 / (2√3 + 6) by rationalizing the denominator with the conjugate of surds. Step-by-step solution provided, covering key concepts like surds, rationalizing denominators, and difference of squares. Suitable for high school students.

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