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Imaginary numbers might sound like a fictional concept, but they're actually a fundamental part of mathematics. The key component here is the imaginary unit "i," which represents the square root of -1. So, whenever you see a negative number under a square root in mathematics, like \(\sqrt{-18}\), you're dealing with an imaginary number. In this particular problem, \(\sqrt{-18}\) becomes \(i\sqrt{18}\). This transformation helps us manipulate and simplify expressions involving square roots of negative numbers.

It's important to remember:

• \(i^2 = -1\)
• \(i^3 = -i\)
• \(i^4 = 1\)

Understanding these fundamental rules will help streamline complex calculations involving imaginary numbers.

In mathematics, especially when dealing with complex numbers, writing an expression in the standard form \(a + bi\) simplifies its interpretation and comparison. Here's how it works:

• "a" is the real part of the complex number.
• "bi" is the imaginary part where "b" is a real number and "i" is the imaginary unit.

In the exercise provided, the expression \(\frac{-5}{11} - \frac{i \sqrt{2}}{11}\) fits the standard form by separating the components into real \(\left(\frac{-5}{11}\right)\) and imaginary parts \(\left(\frac{-\sqrt{2}}{11}i\right)\). Writing numbers this way aids in simplifying mathematical operations and offers clarity in communication between mathematicians.

Performing operations with complex numbers often involves simplifying expressions and ensuring they are presented in a clear format. In the given problem, we begin by transforming the negative square root into an imaginary number, reducing our expression to a format that can be simplified further.

The steps are simple once you break them down:

• Always convert negative square roots using the imaginary unit "i".
• Simplify any radicals left over from square roots.
• Finally, simplify the entire expression such that it adheres to the standard form \(a + bi\).

In the case of \(\frac{-15 - 3i \sqrt{2}}{33}\), the division was performed separately for the real and imaginary parts. Thus, the expression was split into \(\frac{-15}{33}\) and \(\frac{-3i\sqrt{2}}{33}\), leading to the neat result \(\frac{-5}{11} - \frac{\sqrt{2}i}{11}\). This methodical approach ensures the expression is both correct and easy to understand.