Question
\( 1 \leftarrow \quad \) Perform the indicated operations and write the result in standard form. \( -\frac{-8+\sqrt{-20}}{24} \)
Ask by Pritchard Norton. in the United States
Jan 27,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The simplified form of the expression is \( \frac{1}{3} - \frac{\sqrt{5}}{12}i \).
Solution
To simplify the expression and write it in standard form \( a + bi \), follow these steps:
Given expression:
\[
-\frac{-8 + \sqrt{-20}}{24}
\]
1. **Simplify the Square Root of a Negative Number:**
\[
\sqrt{-20} = \sqrt{20} \cdot i = 2\sqrt{5} \, i
\]
2. **Substitute Back into the Expression:**
\[
-\frac{-8 + 2\sqrt{5} \, i}{24}
\]
3. **Distribute the Negative Sign:**
\[
\frac{8 - 2\sqrt{5} \, i}{24}
\]
4. **Separate the Real and Imaginary Parts:**
\[
\frac{8}{24} - \frac{2\sqrt{5}}{24} \, i
\]
5. **Simplify the Fractions:**
\[
\frac{1}{3} - \frac{\sqrt{5}}{12} \, i
\]
**Final Answer:**
\[
\frac{1}{3} - \frac{\sqrt{5}}{12}\,i
\]
Reviewed and approved by the UpStudy tutoring team
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Explain 
Simplify this solution Mind Expander
To simplify the expression \( -\frac{-8+\sqrt{-20}}{24} \), we first tackle \(\sqrt{-20}\). Recall that \(\sqrt{-1} = i\), so: \[ \sqrt{-20} = \sqrt{20} \cdot \sqrt{-1} = \sqrt{20}i = \sqrt{4 \cdot 5}i = 2\sqrt{5}i \] Now substituting back into the expression: \[ -\frac{-8 + 2\sqrt{5}i}{24} \] This simplifies to: \[ \frac{8 - 2\sqrt{5}i}{24} \] Breaking it down further, we can simplify each term: \[ = \frac{8}{24} - \frac{2\sqrt{5}i}{24} = \frac{1}{3} - \frac{\sqrt{5}}{12}i \] Now, in standard form (a + bi): \[ = \frac{1}{3} - \frac{\sqrt{5}}{12}i \] Thus, the final result in standard form is: \[ \frac{1}{3} - \frac{\sqrt{5}}{12}i \]
