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A complex number is formed by adding a real number with an imaginary number. The general form is written as \( a + bi \), where \( a \) and \( b \) are real numbers. The \( i \) represents the imaginary unit, defined as \( \sqrt{-1} \). In this form:

• \( a \) is known as the real part
• \( bi \) is known as the imaginary part

Complex numbers are valuable in mathematics because they provide solutions to equations where real numbers do not suffice, particularly equations involving the square roots of negative numbers.
These numbers have applications in various fields such as engineering, physics, and mathematics itself, helping solve real-world problems.
In essence, any number that can be expressed as a complex number has representations both in a real-world context and in theories that involve imaginary numbers.

Square roots are generally used to find a number which, when multiplied by itself, gives the original number. However, when it comes to negative numbers, things become a bit more interesting.
Since the square of any real number is positive, the square root of a negative number doesn't exist in the set of real numbers. Enter imaginary numbers!
The most fundamental imaginary number is \( i \), representing \( \sqrt{-1} \). Using \( i \), you can express square roots of negative numbers by separating the negative part from the positive. For example:

• To find \( \sqrt{-18} \), separate it into \( \sqrt{-1 \times 18} \).
• This becomes \( \sqrt{-1} \times \sqrt{18} \), which can also be written as \( i \times \sqrt{18} \).

By doing this, we convert a seemingly impossible problem into one that's manageable using complex numbers, as was solved in the original question. It allows us to still perform operations, even when dealing with negative square roots.

The simplification of expressions, particularly those involving complex numbers, plays a crucial role in solving mathematical problems. The ultimate goal is to break down a complex expression into its simplest form without losing its value or equivalence.
The process usually involves several steps:

• Identify any imaginary components, like \( i \), and handle them first.
• Look for ways to break down numbers under the square root, separating into simpler factors if possible. For instance, \( \sqrt{18} = \sqrt{9 \times 2} \), giving \( 3\sqrt{2} \) because \( \sqrt{9} = 3 \).
• Combine these simplified parts back into one expression, ensuring to keep track of the imaginary unit.

Following these steps results in a simpler form that retains the essential characteristics of the original expression.
For example, in the solution given, simplifying \( \sqrt{-18} \) first involved separating \( \sqrt{-1} \) as \( i \). Then \( \sqrt{18} \) was broken down into \( 3\sqrt{2} \), and finally combining them resulted in the expression \( 3i\sqrt{2} \).
This method not only makes the problem manageable but also ensures clarity and precision in solving equations that feature complex numbers.