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When you encounter a square root with a negative number, it signals the involvement of imaginary numbers. For example, in the expression \[\sqrt{-54}\], the negative sign complicates things since you cannot take the square root of a negative number in the real number system. Instead, we introduce the imaginary unit \(i\), defined as \(i = \sqrt{-1}\).
This transforms \(\sqrt{-54}\) into \(\sqrt{54} \cdot i\).
Breaking down \(\sqrt{54}\) involves finding if there are any perfect square factors in 54. By identifying that \(54 = 9 \times 6\), and knowing that 9 is a perfect square \((9 = 3^2)\), the square root can be simplified into \(3\sqrt{6}\).
For the expression \(\sqrt{27}\), recognizing that \(27 = 9 \times 3\), leads to the simplification \(3\sqrt{3}\).
Remember these key ideas:

• Identify any perfect square factors.
• Rewrite the square root using these factors.
• For negative roots, use the imaginary unit \(i\).

Factorization is a critical step in simplifying square roots. It's the process of breaking down a number into basic components that can be multiplied together to yield the original number. In factorization:

• Start by breaking the number into smaller prime factors.
• Look for perfect squares among these factors to simplify calculations.

Let's take \(54\) as an example:
- Break it down into its prime factors: \(54 = 2 \times 3 \times 3 \times 3\), or \(54 = 2 \times 3^3\).
- Identify any squares: \(3^2 = 9\), which is a perfect square.
- Therefore, \(\sqrt{54} = \sqrt{3^2 \times 6} = 3\sqrt{6}\).
Repeating the same for \(27\):
- Break it to prime factors: \(27 = 3 \times 3 \times 3\), or \(27 = 3^3\).
- Perfect square found: \(3^2\), hence \(\sqrt{27} = 3\sqrt{3}\).
Using these steps ensures a cleaner pathway to simplifying expressions.

Complex numbers extend the idea of typical real numbers. They include a part based on the imaginary unit \(i\). A complex number takes the form \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit.
In the expression \(\frac{\sqrt{-54}}{\sqrt{27}}\), after simplification, we end up with \(i\sqrt{2}\). Here's why:
- \(\sqrt{-54}\) involves an imaginary number because of the negative under the root.
- Dividing \(i\sqrt{54}\) by \(\sqrt{27}\) simplifies to \(i\sqrt{2}\).
The result \(i\sqrt{2}\) is a pure imaginary number because it lacks a real component. Complex numbers have unique properties allowing calculations that aren't possible with just real numbers. Always remember:

• Use \(i = \sqrt{-1}\) when simplifying negative square roots.
• Complex numbers bring together real and imaginary parts.

Algebraic manipulation involves rearranging equations and expressions to simplify them or isolate variables. It's a key skill in working with complex numbers and square roots.
In the solution, several algebraic manipulations occur:

• First, rewriting the expression by dealing with the imaginary component \(i\).
• Simplifying square roots by factorization into their basic prime factors.
• Canceling common terms in the numerator and denominator.
• Finally, further simplifying square roots by dividing inside the root.

The step-by-step approach of simplifying \(\frac{3\sqrt{6} \cdot i}{3\sqrt{3}}\) by canceling 3 is a classic example of algebraic manipulation.
Once terms are canceled, the resulting \(\sqrt{2}\cdot i\) showcases the efficient use of algebraic skills to reach a simplified and clean expression. Keeping these principles in mind allows for a much smoother approach to handling complex algebraic problems.