Question
Use the imaginary number \( i \) to rewrite the expression below as a complex number. Simplify all radicals. \( \sqrt{-26} \) i
Ask by Lewis Harris. in the United States
Jan 24,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\( \sqrt{-26} = i\sqrt{26} \)
Solution
To express \( \sqrt{-26} \) as a complex number using the imaginary unit \( i \), follow these steps:
1. **Separate the negative sign:**
\[
\sqrt{-26} = \sqrt{-1 \times 26}
\]
2. **Use the property of square roots:**
\[
\sqrt{-1 \times 26} = \sqrt{-1} \times \sqrt{26}
\]
3. **Replace \( \sqrt{-1} \) with \( i \):**
\[
\sqrt{-1} = i
\]
So,
\[
\sqrt{-26} = i \times \sqrt{26}
\]
4. **Final simplified form:**
\[
\sqrt{-26} = i\sqrt{26}
\]
**Answer:** \( i\sqrt{26} \)
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Simplify this solution Beyond the Answer
To rewrite \( \sqrt{-26} \) using the imaginary unit \( i \), we start by separating the negative sign from the radical: \[ \sqrt{-26} = \sqrt{26} \cdot \sqrt{-1} = \sqrt{26} \cdot i. \] Thus, the expression simplifies to: \[ \sqrt{-26} = i\sqrt{26}. \] So the final answer as a complex number is \( i\sqrt{26} \). Now, don’t forget that complex numbers have real and imaginary parts, so for clarity, you could also express this as \( 0 + i\sqrt{26} \). The real part is 0, and the imaginary part is \( \sqrt{26} \). Whether you're an artist or a scientist, understanding how imaginary numbers work can help unlock creativity in problem-solving!
