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Understanding the standard form of a complex number is foundational to working with these mathematical objects. A complex number is composed of a real part and an imaginary part, and the standard form is written as 'a + bi' where 'a' and 'b' are real numbers, and 'i' is the imaginary unit. The real part is the term without 'i,' and the imaginary part is the term with 'i'.

Simplifying complex expressions often results in a number that can be written in this standard form. For instance, looking at the initial problem \( -6 i^{3}+i^{2} \), once simplified using the properties of 'i,' it becomes \( -1 - 6i \), which aligns with the standard form 'a + bi'. The real part is \( -1 \) and the imaginary part is \( -6i \). Notice that despite the minus sign, the structure still maintains the standard form, which typically places the real part before the imaginary part.

The properties of the imaginary unit 'i' are intriguing and serve as the bedrock for the complex number system. The value of 'i' is defined as the square root of \( -1 \) and does not have a representation on the real number line. Here are a few critical properties of 'i':

• \( i^{2} = -1 \)
• \( i^{3} = i^2 \cdot i = -1 \cdot i = -i \)
• \( i^{4} = (i^{2})^{2} = (-1)^{2} = 1 \), and so on.

These properties are cyclical, meaning that the powers of 'i' repeat every four terms.

When simplifying complex numbers involving 'i', like in the given exercise, these properties help us replace powers of 'i' with corresponding values to make calculations clearer and to eventually rewrite the number in standard form, as shown in the step-by-step solution.

Working with complex numbers entails a variety of operations, including addition, subtraction, multiplication, and division. Each of these operations follows specific rules that ensure the complex numbers are handled properly.

For addition and subtraction, you combine the real parts and combine the imaginary parts separately. For example, if you have two complex numbers \( (a + bi) + (c + di) \), you would add the real parts 'a' and 'c', and then the imaginary parts 'b' and 'd'.

Multiplication and division come with their own set of rules, often requiring the use of the distributive property, conjugates, and the properties of 'i' to simplify expressions. For instance, when multiplying, you would apply the distributive property and combine like terms. In contrast, while dividing, you would multiply by the conjugate of the denominator to eliminate the imaginary parts and simplify further.

Properly applying these operations is essential for simplifying expressions and solving equations that involve complex numbers, as was demonstrated in the provided exercise when substituting the powers of 'i' with their known values.