-8 positive negative square root of -27 all over 6

To simplify the expression \(-8 \pm \sqrt{-27} \, \bigg/ \, 6\):

1. **Simplify the square root of \(-27\)**:
   - \(\sqrt{-27}\) can be written as \(\sqrt{-1 \times 27} = \sqrt{-1} \times \sqrt{27} = i \times \sqrt{27}\).
   - Since \( \sqrt{27} \) simplifies to \( 3\sqrt{3} \), we get \( \sqrt{-27} = 3i\sqrt{3} \).

2. **Rewrite the expression**:
   \[
   \frac{-8 \pm 3i\sqrt{3}}{6}
   \]

3. **Simplify the fraction**:
   - Each term in the numerator can be divided by the denominator:
   \[
   \frac{-8}{6} \pm \frac{3i\sqrt{3}}{6}
   \]
   - Simplify both fractions:
   \[
   -\frac{4}{3} \pm \frac{i\sqrt{3}}{2}
   \]

So, the simplified expression is:
\[
-\frac{4}{3} \pm \frac{i\sqrt{3}}{2}
\]

Would you like more details or have any questions?

Here are 5 related questions for further practice:
1. Simplify \(\frac{6 \pm \sqrt{-50}}{4}\).
2. What are the solutions for the quadratic equation \(x^2 + 4x + 13 = 0\)?
3. Simplify the expression \(\sqrt{-16}\).
4. How would you simplify \(\frac{-3 \pm \sqrt{-12}}{2}\)?
5. Find the roots of the equation \(2x^2 + 3x + 5 = 0\).

**Tip**: When dealing with square roots of negative numbers, remember that \(\sqrt{-1} = i\), which is the imaginary unit.

Learn how to simplify the expression -8 ± √(-27) / 6 step-by-step. This problem involves complex numbers, square roots, and imaginary numbers. Suitable for advanced high school students.

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