91Ó°ÊÓ

The distributive property is a useful mathematical tool that allows you to simplify expressions involving both addition and multiplication. In this context, it helps us operate on complex numbers effectively. When dealing with subtraction, as in the expression \((-3+8i)-(2+3i)\), we use the distributive property to distribute the negative sign across the second set of parentheses. This means we change the signs of all terms inside those parentheses.

For example, applying distributive property, \(-(2+3i) = -2 - 3i\). Therefore, the whole expression becomes \((-3 + 8i) - 2 - 3i\). This step simplifies the expression by removing the parentheses and reversing the signs of the terms inside it. Understanding the distributive property is crucial for simplifying complex expressions as it allows for clear and straightforward manipulation of terms.

Imaginary numbers are a fascinating part of mathematics, extending the concept of number beyond real numbers. The standard form of an imaginary number is expressed using \(i\), the imaginary unit, where \(i\) is defined as the square root of -1. This property gives \(i^2 = -1\). Imaginary numbers are key components in forming complex numbers.

In the exercise \((-3+8i)-(2+3i)\), both complex numbers are made up of a real part and an imaginary part. The \(8i\) and \(3i\) in this expression are imaginary numbers. When simplifying complex numbers, ensure the imaginary numbers are managed according to their rules, treating \(i\) consistently as \(\sqrt{-1}\).

Imaginary numbers are integral in a wide range of applications including engineering, physics, and even in art, where they help represent and solve real-world problems that involve oscillations, waves, and other phenomena.

Combining like terms is a standard algebraic process to simplify expressions. In the realm of complex numbers, this operation is straightforward yet vital. Complex numbers, such as those in our example expression \((-3+8i)-(2+3i)\), consist of real and imaginary components.

To combine like terms, group the real parts and the imaginary parts separately:

• For the real parts: \(-3 - 2 = -5\)
• For the imaginary parts: \(8i - 3i = 5i\)

By combining them correctly, you simplify the expression fully.

The result \(-5 + 5i\) is neatly expressed in the form \(a + bi\), making it much easier to understand and use. Combining like terms not only helps in simplifying the expression but also ensures clear differentiation between real and imaginary components.