In Exercises 1–10, perform the indicated operations and write the result in standard form. (7 + 8i)(7 − 8i)

Verified step by step guidance

1. Recognize that the expression $(7+8i)(7-8i)$ is a product of conjugates.
2. Recall the formula for the product of conjugates: $(a+bi)(a-bi)=a^2+b^2$.
3. Identify $a=7$ and $b=8$ in the expression.
4. Apply the formula: $a^2+b^2=7^2+8^2$.
5. Calculate $7^2$ and $8^2$ separately, then add the results to express the final answer 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 involving them, such as addition, subtraction, multiplication, and division.

Multiplication of Complex Numbers

To multiply complex numbers, you can use the distributive property (also known as the FOIL method for binomials). For two complex numbers (a + bi) and (c + di), the product is calculated as ac + adi + bci + bdi^2. Since i^2 = -1, this simplifies to (ac - bd) + (ad + bc)i, which is crucial for solving the given problem.

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, the result should be simplified to this form for clarity and consistency. In the context of the given exercise, the result of the multiplication should be presented in standard form to complete the operation correctly.