Question
Use the imaginary number \( i \) to rewrite the expression below as a complex number. Simplify all radicals. \( \sqrt{-20} \)
Ask by Herrera Olson. in the United States
Jan 24,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
\( \sqrt{-20} = 2\sqrt{5}\,i \)
Solution
To express \( \sqrt{-20} \) as a complex number using the imaginary unit \( i \), follow these steps:
1. **Separate the Negative Sign:**
\[
\sqrt{-20} = \sqrt{20} \cdot \sqrt{-1}
\]
2. **Simplify \( \sqrt{-1} \) to \( i \):**
\[
\sqrt{-20} = \sqrt{20} \cdot i
\]
3. **Simplify \( \sqrt{20} \):**
\[
\sqrt{20} = \sqrt{4 \times 5} = \sqrt{4} \cdot \sqrt{5} = 2\sqrt{5}
\]
4. **Combine the Results:**
\[
\sqrt{-20} = 2\sqrt{5} \cdot i = 2\sqrt{5}\,i
\]
So, the expression \( \sqrt{-20} \) rewritten as a complex number is:
\[
2\sqrt{5}\,i
\]
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Simplify this solution Beyond the Answer
To rewrite \( \sqrt{-20} \) as a complex number, we start by separating the negative sign. We know that \( \sqrt{-1} \) is represented by the imaginary unit \( i \). Therefore: \[ \sqrt{-20} = \sqrt{20} \cdot \sqrt{-1} = \sqrt{20} \cdot i \] Next, we simplify \( \sqrt{20} \). We can factor \( 20 \) into \( 4 \times 5 \): \[ \sqrt{20} = \sqrt{4 \times 5} = \sqrt{4} \cdot \sqrt{5} = 2\sqrt{5} \] Putting it all together, we have: \[ \sqrt{-20} = 2\sqrt{5} \cdot i \] Thus, the simplified expression is: \[ \sqrt{-20} = 2\sqrt{5} i \]
