In Exercises 9–20, find each product and write the result in standard form.


(3 + 5i)(3 − 5i)

Verified step by step guidance

1. Identify the expression to be multiplied: \((3 + 5i)(3 - 5i)\).
2. Recognize that this is a product of conjugates, which can be simplified using the formula \((a + bi)(a - bi) = a^2 + b^2\).
3. In this expression, \(a = 3\) and \(b = 5\).
4. Apply the formula: \(a^2 + b^2 = 3^2 + 5^2\).
5. Calculate \(3^2\) and \(5^2\) separately, then add the results to express the product in standard form.

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 and 'b' is the coefficient of the imaginary unit 'i', which is 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). For example, when multiplying (a + bi)(c + di), you multiply each part: ac, adi, bci, and bdi^2. Remember that i^2 equals -1, which is crucial for simplifying the result into standard form.

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 simplifying the product of two complex numbers, the goal is to combine like terms and express the result in this standard form, making it easier to interpret and use in further calculations.