$$ 1 \leftarrow \quad $$ Simplify by using the imaginary unit $$ i $$. $$ \frac{-6 \pm \sqrt{-108}}{2} $$

Question

UpStudy AI · Accepted Answer

To simplify the expression $$\frac{-6 \pm \sqrt{-108}}{2}$$ using the imaginary unit $$i$$, follow these steps:

1. **Simplify the Square Root of a Negative Number:**

$$
   \sqrt{-108} = \sqrt{108} \cdot i
   $$
   
   Break down $$108$$ into its prime factors:

$$
   \sqrt{108} = \sqrt{36 \times 3} = \sqrt{36} \times \sqrt{3} = 6\sqrt{3}
   $$
   
   Therefore,

$$
   \sqrt{-108} = 6\sqrt{3} \cdot i
   $$

2. **Substitute Back into the Original Expression:**

$$
   \frac{-6 \pm 6\sqrt{3} \, i}{2}
   $$

3. **Factor Out Common Terms in the Numerator:**

$$
   \frac{6(-1 \pm \sqrt{3} \, i)}{2}
   $$

4. **Simplify the Fraction:**

$$
   \frac{6}{2} (-1 \pm \sqrt{3} \, i) = 3(-1 \pm \sqrt{3} \, i)
   $$

Distribute the 3:

$$
   -3 \pm 3\sqrt{3} \, i
   $$

**Final Simplified Form:**

$$
-3 \pm 3\sqrt{3} \, i
$$
$$-3 \pm 3\sqrt{3} \, i$$

\( 1 \leftarrow \quad \) Simplify by using the imaginary unit \( i \). \( \frac{-6 \pm \sqrt{-108}}{2} \)