Find each product and write the result in standard form. (3 + 5i)(3 - 5i)

Verified step by step guidance

1. Recognize that the expression \((3 + 5i)(3 - 5i)\) is a product of two complex conjugates.
2. Recall the formula for the product of conjugates: \((a + bi)(a - bi) = a^2 + b^2\), where \(a = 3\) and \(b = 5\).
3. Square the real part: calculate \(3^2\).
4. Square the imaginary coefficient: calculate \(5^2\).
5. Add the two results together to write the product in standard form \(a + bi\), noting that the imaginary parts cancel out.

Key Concepts

Complex Numbers

Complex numbers are numbers in the form a + bi, where a and b are real numbers and i is the imaginary unit with the property i² = -1. They extend the real number system and are used to represent quantities that have both real and imaginary parts.

Multiplication of Complex Numbers

To multiply complex numbers, use the distributive property (FOIL method) and apply the rule i² = -1 to simplify. Multiply each term in the first complex number by each term in the second, then combine like terms to get the product.

Standard Form of a Complex Number

The standard form of a complex number is a + bi, where a is the real part and b is the coefficient of the imaginary part. After multiplication, simplify the expression to this form for clarity and consistency.