Simplify by using the imaginary unit $$ i $$ $$ \frac{-8 \pm \sqrt{-128}}{2} $$

Question

UpStudy AI · Accepted Answer

To simplify the expression $$\frac{-8 \pm \sqrt{-128}}{2}$$ using the imaginary unit $$i$$, follow these steps:

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

$$
   \sqrt{-128} = \sqrt{128} \cdot \sqrt{-1} = \sqrt{128} \cdot i
   $$

2. **Simplify $$\sqrt{128}$$:**

$$
   \sqrt{128} = \sqrt{64 \times 2} = \sqrt{64} \cdot \sqrt{2} = 8\sqrt{2}
   $$

So,

$$
   \sqrt{-128} = 8\sqrt{2} \cdot i = 8i\sqrt{2}
   $$

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

$$
   \frac{-8 \pm \sqrt{-128}}{2} = \frac{-8 \pm 8i\sqrt{2}}{2}
   $$

4. **Factor Out the Common Term in the Numerator:**

$$
   \frac{-8 \pm 8i\sqrt{2}}{2} = \frac{8(-1 \pm i\sqrt{2})}{2} = 4(-1 \pm i\sqrt{2})
   $$

5. **Final Simplified Form:**

$$
   -4 \pm 4i\sqrt{2}
   $$

**Therefore, the simplified form is:**

$$
-4 \pm 4\,\sqrt{2}\,i
$$
$$-4 \pm 4\sqrt{2}i$$

Simplify by using the imaginary unit \( i \) \( \frac{-8 \pm \sqrt{-128}}{2} \)