Evaluate the following square root expression.

−8⎯⎯⎯⎯⎯√

The given expression is \( \sqrt{-8} \).

Since we are dealing with the square root of a negative number, the solution involves imaginary numbers. The imaginary unit \( i \) is defined as \( i = \sqrt{-1} \).

Now, we can express \( \sqrt{-8} \) as:

\[
\sqrt{-8} = \sqrt{-1 \times 8} = \sqrt{-1} \times \sqrt{8} = i \times \sqrt{8}
\]

Next, simplify \( \sqrt{8} \). Since \( 8 = 4 \times 2 \), we can write:

\[
\sqrt{8} = \sqrt{4 \times 2} = \sqrt{4} \times \sqrt{2} = 2\sqrt{2}
\]

Thus, the original expression becomes:

\[
\sqrt{-8} = 2i\sqrt{2}
\]

So, the final answer is:

\[
\boxed{2i\sqrt{2}}
\]

Would you like any further details or clarifications on this?

Here are 5 related questions:
1. How do you simplify \( \sqrt{-9} \)?
2. What is the general formula for the square root of a negative number?
3. Can you express \( \sqrt{-16} \) in terms of \( i \)?
4. How do you handle complex numbers in algebraic expressions?
5. What is the value of \( i^2 \)?

**Tip:** Imaginary numbers are essential when dealing with square roots of negative numbers.

Evaluate the following square root expression: −8⎯⎯⎯⎯⎯√

This guide covers the simplification of √(-8), introducing the concept of imaginary numbers and complex numbers. The detailed solution involves breaking down the square root of a negative number, including key formulas such as √(-1) = i and simplifying √8. Ideal for students in Grades 10-12.

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