Question
Jse the imaginary number \( i \) to rewrite the expression below as a complex number. Simplify ill radicals. \( \sqrt{-91} \)
Ask by Pena Conner. in the United States
Jan 24,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\[
\sqrt{-91} = i \sqrt{91}
\]
Solution
To express the square root of a negative number using the imaginary unit \( i \), follow these steps:
1. **Identify the Negative Sign:**
\[
\sqrt{-91} = \sqrt{-1 \times 91}
\]
2. **Use the Property of Imaginary Numbers:**
\[
\sqrt{-1} = i
\]
Therefore,
\[
\sqrt{-91} = \sqrt{-1} \times \sqrt{91} = i \sqrt{91}
\]
3. **Simplify the Radical (if possible):**
- Factorize 91 to check for perfect squares:
\[
91 = 7 \times 13
\]
- Both 7 and 13 are prime numbers, so the square root cannot be simplified further.
**Final Answer:**
\[
\sqrt{-91} = i \sqrt{91}
\]
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Beyond the Answer
To rewrite the expression \( \sqrt{-91} \) as a complex number, we first recognize that the square root of a negative number involves the imaginary unit \( i \), where \( i^2 = -1 \). Thus, we can express \( \sqrt{-91} \) as \( \sqrt{91} \cdot \sqrt{-1} = \sqrt{91} \cdot i \). To simplify, we keep the expression as \( i\sqrt{91} \), as \( \sqrt{91} \) cannot be simplified further, given that 91 does not have any perfect square factors. Therefore, the expression in its simplest form is: \[ \sqrt{-91} = i\sqrt{91} \] Now you've got a neat complex number, just waiting to be used in further mathematical adventures!
