-5 positive negative square root of -10 all over 10.

To simplify the expression \(-5 \pm \sqrt{-10} \, \bigg/ \, 10\):

1. **Simplify the square root of \(-10\)**:
   - \(\sqrt{-10}\) can be written as \(\sqrt{-1 \times 10} = \sqrt{-1} \times \sqrt{10} = i \times \sqrt{10}\).

2. **Rewrite the expression**:
   \[
   \frac{-5 \pm i\sqrt{10}}{10}
   \]

3. **Simplify the fraction**:
   - Each term in the numerator can be divided by the denominator:
   \[
   \frac{-5}{10} \pm \frac{i\sqrt{10}}{10}
   \]
   - Simplify both fractions:
   \[
   -\frac{1}{2} \pm \frac{i\sqrt{10}}{10}
   \]

So, the simplified expression is:
\[
-\frac{1}{2} \pm \frac{i\sqrt{10}}{10}
\]

Would you like more details or have any questions?

Here are 5 related questions for further practice:
1. Simplify \(\frac{7 \pm \sqrt{-24}}{12}\).
2. Solve the equation \(x^2 - 6x + 10 = 0\) using the quadratic formula.
3. How do you simplify \(\frac{-3 \pm \sqrt{-18}}{9}\)?
4. Simplify the expression \(\sqrt{-20}\).
5. What are the roots of the equation \(x^2 + 2x + 5 = 0\)?

**Tip**: Always look to factor or simplify fractions when working with complex numbers in the numerator.

Learn how to simplify the expression -5 ± √(-10) / 10 with detailed steps and explanations. This problem involves complex numbers, square roots, and the quadratic formula. Suitable for high school students studying advanced algebra.

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