Question
Use the imaginary number \( i \) to rewrite the expression below as a complex number. Simplify all radicals. \( \sqrt{-74} \) \begin{tabular}{|c|c}\hline \\ \hline\end{tabular}\( \sqrt{ } \)
Ask by Washington Simmons. in the United States
Jan 24,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\[
\sqrt{-74} = \sqrt{74}\,i
\]
Solution
To rewrite \( \sqrt{-74} \) as a complex number using the imaginary unit \( i \), follow these steps:
1. **Identify the Negative Sign:**
The square root of a negative number can be expressed using \( i \), where \( i = \sqrt{-1} \).
2. **Separate the Radicand:**
\[
\sqrt{-74} = \sqrt{74 \times -1} = \sqrt{74} \times \sqrt{-1}
\]
3. **Apply the Definition of \( i \):**
\[
\sqrt{-1} = i
\]
4. **Combine the Terms:**
\[
\sqrt{-74} = \sqrt{74} \times i = \sqrt{74}\,i
\]
5. **Simplify the Radical (if possible):**
- Check if 74 can be factored into a perfect square and another factor.
- \( 74 = 2 \times 37 \)
- Neither 2 nor 37 are perfect squares, so \( \sqrt{74} \) cannot be simplified further.
**Final Answer:**
\[
\sqrt{-74} = \sqrt{74}\,i
\]
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Bonus Knowledge
To rewrite the expression \( \sqrt{-74} \) as a complex number, we can utilize the imaginary unit \( i \), where \( i = \sqrt{-1} \). By incorporating \( i \), we have: \[ \sqrt{-74} = \sqrt{74} \cdot \sqrt{-1} = \sqrt{74} \cdot i \] Next, we can perform a bit of simplification. The expression is now in the form of \( a + bi \), where here \( a = 0 \) and \( b = \sqrt{74} \). Thus, the simplified form of \( \sqrt{-74} \) as a complex number is: \[ 0 + \sqrt{74} i \] or simply: \[ \sqrt{74} i \]
