Question
Write in terms of \( i \). Simplify your answer as much as possible \[ \sqrt{-75} \]
Ask by Simpson Herrera. in the United States
Jan 24,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\[
\sqrt{-75} = 5\sqrt{3}\,i
\]
Solution
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:**
\[
\sqrt{-75} = 5\sqrt{3}\,i
\]
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Extra Insights
To simplify \(\sqrt{-75}\), we start by recognizing that we can write it in terms of \(i\), where \(i = \sqrt{-1}\). We can express \(-75\) as \(-1 \times 75\). Therefore, we have: \[ \sqrt{-75} = \sqrt{-1 \times 75} = \sqrt{-1} \times \sqrt{75} = i \sqrt{75} \] Next, we simplify \(\sqrt{75}\). We can factor \(75\) into \(25 \times 3\): \[ \sqrt{75} = \sqrt{25 \times 3} = \sqrt{25} \times \sqrt{3} = 5\sqrt{3} \] Now, substitute back into the expression: \[ \sqrt{-75} = i \cdot 5\sqrt{3} = 5i\sqrt{3} \] Thus, the simplified form of \(\sqrt{-75}\) is: \[ 5i\sqrt{3} \]
