Question
Use the imaginary number \( i \) to rewrite the expression below as a complex number. Simplify all radicals. \( \sqrt{-33} \)
Ask by Wheeler Schofield. in the United States
Feb 03,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\[
\sqrt{33}\,i
\]
Solution
To rewrite the expression \( \sqrt{-33} \) as a complex number using the imaginary unit \( i \), follow these steps:
1. **Understand the Imaginary Unit**:
- The imaginary unit \( i \) is defined as \( i = \sqrt{-1} \).
2. **Rewrite the Square Root of a Negative Number**:
- \( \sqrt{-33} \) can be expressed as \( \sqrt{33} \times \sqrt{-1} \).
- Using the definition of \( i \), this becomes \( \sqrt{33} \times i \).
3. **Simplify**:
- The expression simplifies to \( \sqrt{33}\,i \).
**Final Answer:**
\[
\sqrt{33}\,i
\]
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Simplify this solution Bonus Knowledge
To rewrite \( \sqrt{-33} \) as a complex number, we start by factoring out the negative sign. We know that \( \sqrt{-1} = i \), so we can express \( \sqrt{-33} \) as: \[ \sqrt{-33} = \sqrt{33} \cdot \sqrt{-1} = \sqrt{33} \cdot i \] Thus, the simplified expression for \( \sqrt{-33} \) in terms of the imaginary unit \( i \) is: \[ \sqrt{-33} = i \sqrt{33} \]
