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The imaginary unit, denoted as \(i\), is a fundamental concept in complex numbers. It is defined as the square root of -1, which is written as \(i^2 = -1\). This definition allows us to solve equations that involve negative square roots, thus opening the door to a broader number system beyond real numbers. Using \(i\), we can express numbers like \(\sqrt{-4}\) as \(2i\). When working with complex numbers, \(i\) plays a crucial role in calculations, particularly when squaring and simplifying expressions.
Whenever you see \(i^2\), you can replace it with -1, as shown in the solution. This small adjustment can make calculations much more straightforward. For example, transforming the product of two imaginary numbers into a real number, as demonstrated with the product \(-3i \times 7i\), which simplifies to \(21 \times i^2 = -21\). Remembering this core property of \(i\) will streamline your computation processes.

The distributive property is a fundamental algebraic principle that helps in simplifying expressions involving multiplication over addition or subtraction. This means you "distribute" by multiplying each term inside a bracket by the number or expression outside. Mathematically, this property is expressed as: \(a(b + c) = ab + ac\).
In the context of our exercise, the distributive property is applied to multiply \(-3i\) by both terms inside the parentheses \((7i - 5)\). This results in two separate multiplication operations: \(-3i \times 7i\) and \(-3i \times -5\).

• First, you calculate \(-3i \times 7i\), which simplifies using the imaginary unit properties described.
• Next, \(-3i \times -5\) results in a straightforward multiplication, yielding \(15i\).

The distributive property simplifies handling expressions in complex numbers, making them easier to manage and solve.

Complex numbers are typically expressed in standard form as \(a + bi\), where \(a\) and \(b\) are real numbers, and \(i\) is the imaginary unit. This form provides a clear way to separate the real part \(a\) from the imaginary part \(bi\).
In the solution, after applying the distributive property and simplifying the expression, we have \(-21 + 15i\). This is already in the standard form of a complex number, where:

• \(-21\) is the real component.
• \(15i\) is the imaginary component.

This standard representation allows easy addition, subtraction, and comparison of complex numbers, making operations straightforward. When working through problems involving complex numbers, strive to express your solution in this standard form.