Question
Simplify, if possible \( \sqrt{-18}=\square \) (Simplify your answer. Type an exact answer, using radicals and \( i \) nee
Ask by Peterson Weber. in the United States
Jan 27,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\( \sqrt{-18} = 3\sqrt{2}\,i \)
Solution
To simplify \( \sqrt{-18} \), we'll use the concept of imaginary numbers, where \( i = \sqrt{-1} \).
\[
\sqrt{-18} = \sqrt{-1 \times 18} = \sqrt{-1} \times \sqrt{18} = i \times \sqrt{18}
\]
Next, simplify \( \sqrt{18} \):
\[
\sqrt{18} = \sqrt{9 \times 2} = \sqrt{9} \times \sqrt{2} = 3\sqrt{2}
\]
Now, combine the results:
\[
\sqrt{-18} = i \times 3\sqrt{2} = 3\sqrt{2}\,i
\]
**Final Answer:**
\[
3\sqrt{2}\,i
\]
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Simplify this solution Bonus Knowledge
To simplify \( \sqrt{-18} \), we first take out the negative sign using \( i \), the imaginary unit. This gives us: \[ \sqrt{-18} = \sqrt{18} \cdot \sqrt{-1} = \sqrt{18} \cdot i \] Next, we simplify \( \sqrt{18} \): \[ \sqrt{18} = \sqrt{9 \cdot 2} = \sqrt{9} \cdot \sqrt{2} = 3\sqrt{2} \] Combining these results, we get: \[ \sqrt{-18} = 3\sqrt{2} \cdot i \] So, the final answer is: \[ \sqrt{-18} = 3\sqrt{2}i \]
