Find each product. Write answers in standard form. i(3-4i)(3+4i)

Verified step by step guidance

1. Step 1: Distribute the 'i' to each term in the first parentheses to get '3i - 4i^2'.
2. Step 2: Remember that 'i^2' is equal to '-1'. So, replace 'i^2' with '-1' to get '3i - 4(-1)' or '3i + 4'.
3. Step 3: Now, multiply '(3i + 4)' with '(3 + 4i)'. First, distribute '3i' to each term in the second parentheses to get '9i + 12i^2'.
4. Step 4: Then, distribute '4' to each term in the second parentheses to get '12 + 16i'.
5. Step 5: Combine the results from steps 3 and 4 to get '9i + 12i^2 + 12 + 16i'. Again, replace 'i^2' with '-1' to get the final expression 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) and combine like terms. When multiplying two complex numbers, such as (a + bi)(c + di), the result is ac + adi + bci + bdi^2. Since i^2 equals -1, this simplifies to (ac - bd) + (ad + bc)i, which is crucial for finding the product in the given question.

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 solving problems involving complex numbers, it is important to present the final answer in this form to clearly indicate the real and imaginary components. This format helps in further calculations and understanding the properties of complex numbers.