Write in terms of $$ i $$. Simplify your answer as much as possible $$ \sqrt{-75} $$

Question

UpStudy AI · Accepted Answer

To express $$\sqrt{-75}$$ in terms of the imaginary unit $$i$$ and simplify it, follow these steps:

1. **Factor the negative inside the square root:**
   $$
   \sqrt{-75} = \sqrt{75 \times -1}
   $$

2. **Separate the square root of the positive number and the imaginary unit:**
   $$
   \sqrt{75 \times -1} = \sqrt{75} \times \sqrt{-1}
   $$

3. **Recognize that $$\sqrt{-1}$$ is defined as $$i$$:**
   $$
   \sqrt{75} \times \sqrt{-1} = \sqrt{75} \times i
   $$

4. **Simplify $$\sqrt{75}$$ by factoring out perfect squares:**
   $$
   \sqrt{75} = \sqrt{25 \times 3} = \sqrt{25} \times \sqrt{3} = 5\sqrt{3}
   $$

5. **Combine the simplified terms:**
   $$
   5\sqrt{3} \times i = 5\sqrt{3}i
   $$

**Final Answer:**
$$
\sqrt{-75} = 5\sqrt{3}\,i
$$
$$ \sqrt{-75} = 5\sqrt{3}\,i $$

Write in terms of \( i \). Simplify your answer as much as possible \[ \sqrt{-75} \]