How do you simplify the following expression? [(2 ± (-12)^(1/2))/2]

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 \(i = \sqrt{-1}\).

Learn how to simplify the expression (2 ± √(-12))/2 using imaginary numbers. This step-by-step guide explains key concepts in complex numbers and how to handle square roots of negative numbers. Ideal for high school students in Grades 9-12.

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