Simplify the following radical.

−27⎯⎯⎯⎯⎯⎯⎯√−27

To simplify the expression \(\sqrt{-27}\), we need to deal with the negative sign inside the square root, which involves using imaginary numbers.

Here's the process:

\[
\sqrt{-27} = \sqrt{-1 \times 27} = \sqrt{-1} \times \sqrt{27}
\]

Since \(\sqrt{-1} = i\), where \(i\) is the imaginary unit, the expression becomes:

\[
\sqrt{-27} = i \times \sqrt{27}
\]

Now, simplify \(\sqrt{27}\):

\[
\sqrt{27} = \sqrt{9 \times 3} = \sqrt{9} \times \sqrt{3} = 3\sqrt{3}
\]

So, the simplified form of \(\sqrt{-27}\) is:

\[
\sqrt{-27} = 3i\sqrt{3}
\]

Thus, the final simplified expression is:

\[
\boxed{3i\sqrt{3}}
\]

Would you like more details or have any questions? Here are 5 related questions for further practice:

1. Simplify \(\sqrt{-50}\).
2. Simplify \(\sqrt{-72}\).
3. What is the value of \(i^2\)?
4. Simplify \(\sqrt{-8}\).
5. Express \(\sqrt{-18}\) in its simplest form.

**Tip:** When simplifying radicals with negative numbers, always remember to factor out \(\sqrt{-1}\) as \(i\), the imaginary unit!

Simplify the following radical: \(\sqrt{-27}\)

Learn how to simplify the radical expression \(\sqrt{-27}\), dealing with imaginary numbers and square roots. This problem explores imaginary numbers and the process of simplifying radicals. Perfect for students in Grades 9-12 studying complex numbers.

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