In Exercises 48–57, perform the indicated operations and write the result in standard form. (7-i)(2+3i)

Key Concepts

Complex Numbers

Complex numbers are numbers that have a real part and an imaginary part, expressed in the form a + bi, where a is the real part, b is the imaginary part, and i is the imaginary unit defined as the square root of -1. Understanding complex numbers is essential for performing operations such as addition, subtraction, multiplication, and division.

Multiplication of Complex Numbers

To multiply complex numbers, you apply the distributive property (also known as the FOIL method for binomials) and combine like terms. For example, when multiplying (7 - i)(2 + 3i), you multiply each term in the first complex number by each term in the second, and then simplify the result by combining real and imaginary parts.

Standard Form of Complex Numbers

The standard form of a complex number is a + bi, where a and b are real numbers. After performing operations on complex numbers, it is important to express the result in this form, ensuring that the real and imaginary components are clearly separated, which aids in further calculations and interpretations.