Question
Use the imaginary number \( i \) to rewrite the expression below as a complex number. Simplify all radicals. \[ -20+\sqrt{-8} \]
Ask by Flynn Watkins. in the United States
Jan 24,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\(-20 + 2\sqrt{2}i\)
Solution
To rewrite the expression \(-20 + \sqrt{-8}\) as a complex number using the imaginary unit \(i\), follow these steps:
1. **Express \(\sqrt{-8}\) in terms of \(i\):**
\[
\sqrt{-8} = \sqrt{8} \cdot \sqrt{-1} = \sqrt{8} \cdot i
\]
2. **Simplify \(\sqrt{8}\):**
\[
\sqrt{8} = \sqrt{4 \times 2} = \sqrt{4} \cdot \sqrt{2} = 2\sqrt{2}
\]
3. **Substitute back into the expression:**
\[
-20 + \sqrt{-8} = -20 + 2\sqrt{2} \cdot i
\]
So, the expression rewritten as a complex number is:
\[
-20 + 2\sqrt{2}\,i
\]
**Final Answer:**
\[-20 + 2\,\sqrt{2}\;i\]
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution Bonus Knowledge
To rewrite the expression \( -20+\sqrt{-8} \) as a complex number, we start by simplifying \( \sqrt{-8} \). We can express \( \sqrt{-8} \) as: \[ \sqrt{-8} = \sqrt{8} \cdot \sqrt{-1} = \sqrt{8} \cdot i \] Next, we simplify \( \sqrt{8} \): \[ \sqrt{8} = \sqrt{4 \cdot 2} = \sqrt{4} \cdot \sqrt{2} = 2\sqrt{2} \] So, we have: \[ \sqrt{-8} = 2\sqrt{2} \cdot i \] Now, we can substitute back into the original expression: \[ -20 + \sqrt{-8} = -20 + 2\sqrt{2}i \] Thus, the expression simplified as a complex number is: \[ -20 + 2\sqrt{2}i \]
