Perform each operation. Write answers in standard form. (5-11i)(5+11i)

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 involving them, 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 (a + bi)(c + di), you calculate ac, adi, bci, and bdi^2, remembering that i^2 = -1. This process results in a new complex number in the form of (ac - bd) + (ad + bc)i.

Standard Form of Complex Numbers

The standard form of a complex number is expressed as a + bi, where a and b are real numbers. When performing operations on complex numbers, it is important to simplify the result into this form for clarity and consistency. This involves combining real and imaginary parts and ensuring that any instances of i^2 are replaced with -1.