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In complex numbers, the standard form is a way to neatly express a combination of real and imaginary parts. This form is written as \(a + bi\), where:

• \(a\) is the real part, a regular number without any imaginary component.
• \(b\) is the coefficient of the imaginary unit \(i\), representing the imaginary part.

In expressions involving complex numbers, such as the one given here, \(-96 - 36i\),you can see it takes the shape of a standard form complex number:

• \(-96\) is the real component \(a\), and
• \(-36i\) is the imaginary component \(bi\).

Using the standard form makes it easier to identify and work with complex numbers in equations and calculations.This form is particularly useful since it clarifies the separation between the real and imaginary elements.Keep in mind that the real number \(a\) and the imaginary part \(bi\) are placed to give a clear structure for further mathematical operations on complex numbers.

The distributive property is a fundamental mathematical principle used to simplify expressions when an operation is applied to a group of terms within parentheses.In its most basic form, it states that:\[a(b + c) = ab + ac\]This means multiplying \(a\) by every term inside the parentheses individually and then summing the results.Let's look at how this works in the original exercise:

• The expression \(-12(8 + 3i)\)is distributed by multiplying \(-12\) by \(8\) and also by \(3i\).
• As a result, \(-12 \times 8 = -96\)and \(-12 \times 3i = -36i\).

Using the distributive property allows for simplification and helps us work step by step in breaking down complex expressions.By applying this property correctly, it ensures every component in the parentheses is accounted for, making calculations more accurate and manageable.

An imaginary unit is a mathematical concept used to extend the real number system. It is represented by the symbol \(i\) and is defined with an important property:

• \(i = \sqrt{-1}\)

The introduction of the imaginary unit facilitates calculations involving the square roots of negative numbers, which lack solutions among the real numbers.For example, if we use \(i\) in a multiplication such as \(-12 \times 3i\),unlike a real number, this multiplies by the imaginary part:

• Resulting in \(-36i\).

Imaginary numbers, thus, enable us to solve and work with equations that cross the bounds of possible real solutions.Use of \(i\)opens up new possibilities in mathematics, particularly in fields like engineering and physics where complex numbers are often used to solve real-world problems.Additionally, it's crucial to keep the imaginary unit distinct from the real number in solutions, expressed in the standard form as \(a + bi\), to ensure comprehension of each component's role in complex calculations.