To simplify the expression $\left( \frac{2 \pm \sqrt{-12}}{2} \right)$, follow these steps:

Step 1: Simplify the square root of $-12$

Since $\sqrt{-12}$ involves a negative number, we need to express this in terms of imaginary numbers: $\sqrt{-12} = \sqrt{12} \cdot i = 2\sqrt{3} \cdot i$ Here, $i$ represents the imaginary unit, where $i^2 = -1$.

Step 2: Substitute $\sqrt{-12}$ back into the expression

Now the expression becomes: $\frac{2 \pm 2\sqrt{3}i}{2}$

Step 3: Simplify the fraction

You can now simplify by dividing both terms in the numerator by 2: $\frac{2}{2} \pm \frac{2\sqrt{3}i}{2} = 1 \pm \sqrt{3}i$

Final Simplified Expression:

The simplified expression is: $1 \pm \sqrt{3}i$

Let me know if you have questions or need further details!

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Here are 5 related questions:

1. How do you simplify other complex numbers involving square roots?
2. What is the significance of the imaginary unit $i$?
3. How do you plot complex numbers on the complex plane?
4. How do you add and subtract complex numbers?
5. What are the polar coordinates of the complex number $1 + \sqrt{3}i$?

Tip: Whenever you encounter negative square roots, think of imaginary numbers using $i = \sqrt{-1}$.