Question
Use the imaginary number \( i \) to rewrite the expression below as a complex number. Simplify all radicals. \[ 2+\sqrt{-18} \]
Ask by Nunez Herrera. in the United States
Jan 24,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The expression \(2 + \sqrt{-18}\) as a complex number is \(2 + 3\sqrt{2}i\).
Solution
To rewrite the expression \( 2 + \sqrt{-18} \) as a complex number, follow these steps:
1. **Express the Square Root of a Negative Number:**
The square root of a negative number can be expressed using the imaginary unit \( i \), where \( i = \sqrt{-1} \).
\[
\sqrt{-18} = \sqrt{18} \cdot \sqrt{-1} = \sqrt{18} \cdot i
\]
2. **Simplify the Radical:**
Simplify \( \sqrt{18} \) by factoring out the perfect square.
\[
\sqrt{18} = \sqrt{9 \times 2} = \sqrt{9} \cdot \sqrt{2} = 3\sqrt{2}
\]
Therefore,
\[
\sqrt{-18} = 3\sqrt{2} \cdot i = 3\sqrt{2}i
\]
3. **Combine with the Real Part:**
Now, substitute back into the original expression:
\[
2 + \sqrt{-18} = 2 + 3\sqrt{2}i
\]
So, the expression \( 2 + \sqrt{-18} \) rewritten as a complex number is:
\[
2 + 3\sqrt{2}\,i
\]
Answer: \(2 + 3\,\sqrt{2}\,i\)
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Extra Insights
To rewrite the expression \( 2+\sqrt{-18} \) as a complex number, we first need to recognize that \( \sqrt{-18} \) can be expressed using the imaginary unit \( i \). We can rewrite it as: \[ \sqrt{-18} = \sqrt{18} \cdot \sqrt{-1} = \sqrt{18} \cdot i \] Next, we simplify \( \sqrt{18} \): \[ \sqrt{18} = \sqrt{9 \cdot 2} = \sqrt{9} \cdot \sqrt{2} = 3\sqrt{2} \] Now substituting this back into our expression, we have: \[ \sqrt{-18} = 3\sqrt{2} \cdot i \] So our initial expression now becomes: \[ 2 + \sqrt{-18} = 2 + 3\sqrt{2}i \] Thus, the final expression as a complex number is: \[ 2 + 3\sqrt{2}i \]
