Simplify the following expression: [(2 ± (-12)^(1/2))/2]

We are tasked with simplifying the expression:

\[
\frac{2 \pm \sqrt{-12}}{2}
\]

### Step 1: Simplify \(\sqrt{-12}\)
We know that \(\sqrt{-12}\) involves a negative number under the square root, which introduces the imaginary unit \(i\), where \(i = \sqrt{-1}\). Thus, we can rewrite:

\[
\sqrt{-12} = \sqrt{12} \cdot i
\]

Since \(\sqrt{12} = 2\sqrt{3}\), we have:

\[
\sqrt{-12} = 2\sqrt{3} \cdot i = 2i\sqrt{3}
\]

### Step 2: Substitute into the original expression
Now, substitute \(\sqrt{-12} = 2i\sqrt{3}\) back into the expression:

\[
\frac{2 \pm 2i\sqrt{3}}{2}
\]

### Step 3: Simplify the fraction
We can simplify this expression by dividing both terms in the numerator by 2:

\[
\frac{2}{2} \pm \frac{2i\sqrt{3}}{2} = 1 \pm i\sqrt{3}
\]

### Final Answer:
The simplified form of the expression is:

\[
1 \pm i\sqrt{3}
\]

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Do you want more details on any step or have any questions?

Here are 5 related questions:
1. How do you simplify other expressions with square roots of negative numbers?
2. What is the significance of the imaginary unit \(i\)?
3. How would the simplification change if \(-12\) were positive?
4. Can this result be represented graphically in the complex plane?
5. How do you simplify more complex expressions involving imaginary numbers?

**Tip:** When simplifying expressions with imaginary numbers, always factor out \(i\) to handle negative square roots properly.

Learn how to simplify the expression [(2 ± (-12)^(1/2))/2] involving imaginary numbers. This step-by-step solution explores key concepts like square roots of negative numbers, complex number simplification, and the imaginary unit i. Perfect for students in Grades 10-12.

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