91Ó°ÊÓ

The imaginary unit, represented as \( i \), is a fundamental concept in complex numbers. It's an essential element because it enables us to extend the real number system to include roots of negative numbers. The defining property of \( i \) is that its square equals \( -1 \), expressed as:

• \( i^2 = -1 \)

When dealing with multiplication involving \( i \), it is important to remember its unique properties. For example, \( i^3 = i imes i^2 = i imes (-1) = -i \) and \( i^4 = i^2 imes i^2 = (-1) imes (-1) = 1 \).
These cyclical powers of \( i \) simplify computations involving imaginary numbers and appear often when solving problems like the one in the exercise. Knowing these properties helps streamline complex calculations.

In complex number multiplication, it is crucial to remember the special behavior of the imaginary unit \( i \). When multiplying complex numbers, follow the distributive property and apply the fundamental property of \( i^2 = -1 \).
Let's break down the multiplication process using the exercise:

• First multiply two imaginary terms: \( -6i \times -i = 6i^2 \). As discussed earlier, replace \( i^2 \) with \(-1\), simplifying it to \(-6\).
• Next, multiply the result \(-6\) with the third term \( 6i \):\(-6 \times 6i = -36i \).

Approaching the multiplication step by step prevents mistakes and ensures that each imaginary unit is properly accounted for. Remember to convert every \( i^2 \) to \(-1\) to simplify calculations without confusion.

The standard form of a complex number is expressed as \( a + bi \), where \( a \) and \( b \) are real numbers, and \( i \) is the imaginary unit. To express a complex number in standard form means writing it in this form so that real and imaginary parts are distinct.
In the context of the exercise, the result of the multiplication was \(-36i\). In this case, the real part \( a = 0 \) and the imaginary part \( b = -36 \). Thus, the number is expressed as \( 0 - 36i \).
Standard form makes it easy to identify real and imaginary components while performing operations like addition, subtraction, or further multiplication. This clarity is vital for systematic problem-solving and understanding the behavior of complex numbers in various algebraic structures.