Question
Use the imaginary number i to rewrite the expression below as a complex number. Simplify all radicals. \( \sqrt{-35} \)
Ask by Ruiz Nichols. in the United States
Jan 24,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\( \sqrt{35}\,i \)
Solution
To express \( \sqrt{-35} \) as a complex number using the imaginary unit \( i \), follow these steps:
1. **Break Down the Radical:**
\[
\sqrt{-35} = \sqrt{35 \times (-1)}
\]
2. **Separate the Square Roots:**
\[
\sqrt{35 \times (-1)} = \sqrt{35} \times \sqrt{-1}
\]
3. **Use the Definition of \( i \):**
\[
\sqrt{-1} = i
\]
So,
\[
\sqrt{35} \times \sqrt{-1} = \sqrt{35} \times i = \sqrt{35}i
\]
4. **Final Expression:**
\[
\sqrt{-35} = \sqrt{35}i
\]
This is the simplified form of the complex number.
**Answer:** \( \sqrt{35}\,i \)
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Simplify this solution Extra Insights
The expression \( \sqrt{-35} \) can be rewritten using the imaginary unit \( i \) as \( \sqrt{35} \cdot i \). This is because the square root of a negative number introduces \( i \), which represents the square root of -1. Therefore, the final expression simplifies to \( i\sqrt{35} \). For those venturing into the world of complex numbers, remember that every time you see a negative under a square root, simply pull out an \( i \)! It’s like having a handy tool for navigating through the tricky terrain of square roots, making math much more fun—just like diving into a hidden treasure chest beneath the sea!
