Question
se the imaginary number \( i \) to rewrite the expression below as a complex number. Simplify \( \sqrt{-39} \)
Ask by Vaughan Long. in the United States
Jan 24,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\[
\sqrt{-39} = \sqrt{39}\,i
\]
Solution
To rewrite the expression \( \sqrt{-39} \) as a complex number using the imaginary unit \( i \), follow these steps:
1. **Understand the Imaginary Unit \( i \):**
The imaginary unit \( i \) is defined as:
\[
i = \sqrt{-1}
\]
2. **Rewrite \( \sqrt{-39} \):**
Use the property of square roots involving negative numbers:
\[
\sqrt{-39} = \sqrt{39} \cdot \sqrt{-1} = \sqrt{39} \cdot i
\]
3. **Express as a Complex Number:**
A complex number is generally written in the form \( a + bi \), where \( a \) is the real part and \( b \) is the imaginary part. In this case, there is no real part, so \( a = 0 \):
\[
\sqrt{-39} = 0 + \sqrt{39}i
\]
However, it's common to simply write it as:
\[
\sqrt{39}i
\]
**Final Answer:**
\[
\sqrt{-39} = \sqrt{39}\,i
\]
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Extra Insights
To rewrite \( \sqrt{-39} \) using the imaginary unit \( i \), you can express it as follows: \[ \sqrt{-39} = \sqrt{39} \cdot \sqrt{-1} = \sqrt{39} \cdot i = i\sqrt{39}. \] Thus, the expression simplifies to \( i\sqrt{39} \), where \( i \) represents the square root of -1, turning this initially troublesome square root into a manageable complex number. If you need to perform operations with this complex number, remember that multiplying or dividing by \( i \) can be a fun twist! For example, multiplying \( i\sqrt{39} \) by \( i \) results in \( -39 \), converting it back to a real number. The interplay between real and imaginary numbers opens up new possibilities in mathematics!
