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Complex multiplication is a method used to multiply two complex numbers. A complex number is typically represented in the form \(a + bi\), where \(a\) is the real part and \(bi\) is the imaginary part with \(i\) representing the imaginary unit.

When multiplying complex numbers, you can use the distributive property, much like multiplying binomials. Here's the process:

• Multiply each term in the first complex number by each term in the second complex number.
• Combine like terms.
• Simplify using the property \(i^2 = -1\).

This means that when you have two complex numbers, \((a + bi)\) and \((c + di)\), you expand them to \(ac + adi + bci + bdi^2\).

Afterward, simplify using \(i^2 = -1\) to get your final result in \(a + bi\) form. This process helps us evaluate complex expressions like in the exercise.

The imaginary unit, denoted as \(i\), is a fundamental concept in complex numbers. It is defined as the square root of -1, so \(i = \sqrt{-1}\).

This definition leads to the important property \(i^2 = -1\). This property is crucial when simplifying expressions involving squares of imaginary units, as it allows us to transform these into real numbers. For instance, when multiplying complex numbers, you might encounter terms like \(i^2\), and you would replace it with -1.

Understanding \(i\) is vital because it allows us to work with negative square roots, something we can't do with real numbers. In the exercise provided, recognizing that \(\sqrt{-5}\) should be written as \(i\sqrt{5}\) helps in transforming the equation into a solvable form.

The \(a + bi\) form, also known as standard form, is a way to express complex numbers. Here, \(a\) represents the real part and \(bi\) represents the imaginary part of the number.

When simplifying expressions involving complex numbers, the goal is to separate the real and imaginary components such that the expression is clearly presented in the \(a + bi\) form. This requires recognizing which parts of your expression are real and which are imaginary.

• The real part is comprised of all numbers not multiplied by \(i\).
• The imaginary part includes terms multiplied by \(i\).

In solving the exercise, we separate terms like \(-16 - \sqrt{35}\) as the real part and \((8\sqrt{7} + 2\sqrt{5})i\) as the imaginary part. This allows us to write the final answer neatly in the \(a + bi\) format, making it understandable and neatly organized.