91Ó°ÊÓ

The distributive property is a fundamental concept in algebra that allows you to simplify expressions and perform multiplication efficiently. It states that a term outside the parentheses can be multiplied with each term inside the parentheses, and then the results are added together. For example, if you have an expression like \(a(b + c)\), using the distributive property means you'll expand it to \(ab + ac\). This property helps in breaking down complex multiplications into simpler, more manageable steps.

In our exercise, we applied this property to distribute \-3i\ across both terms inside the parentheses: \(7i - 5\). It simplifies \-3i(7i)\ and \-3i(-5)\ to get the expression \-21 - 15i\. Using this method ensures that each term is multiplied appropriately, and it sets the stage for further simplifications, especially when dealing with complex numbers.

Imaginary numbers are an extension of the real numbers used to solve equations that don't have solutions among real numbers. The basic imaginary unit is represented as \(i\), which is defined as the square root of \(-1\). Thus, the equation \(i^2 = -1\) is fundamental in working with these numbers.

When multiplied, as in \(-3i * 7i\), these imaginary units use the property \(i^2 = -1\) to convert into real numbers. In this step, \(-3i * 7i\) turns into \21i^2\, which further simplifies to \-21\, a real number since \(i^2\) converts to \-1\. Imaginary numbers are often paired with real numbers to form complex numbers, which helps solve a variety of mathematical problems that extend beyond the realm of real numbers.

The standard form of a complex number is expressed as \(a + bi\), where \(a\) stands for the real part and \(bi\) represents the imaginary part. This format is particularly helpful because it allows you to easily identify the real and imaginary components of complex numbers, which are essential in complex arithmetic.

In our exercise, the final outcome after applying the distributive property is \-21 - 15i\. Here, \-21\ is the real part, while \-15i\ is the imaginary part, making it neatly arranged in the standard form. Displaying complex numbers in this form ensures clarity and precision when performing further calculations or when communicating results in complex arithmetic.