In Exercises 9–20, find each product and write the result in standard form. (7 − 5i)(−2 − 3i)

Verified step by step guidance

1. Identify the expression to be expanded: $(7-5i)(-2-3i)$.
2. Apply the distributive property (also known as the FOIL method for binomials) to expand the expression: $(7)(-2)+(7)(-3i)+(-5i)(-2)+(-5i)(-3i)$.
3. Calculate each term separately: $7\times -2$, $7\times -3i$, $-5i\times -2$, and $-5i\times -3i$.
4. Combine the real parts and the imaginary parts separately. Remember that $i^2=-1$.
5. Write the final expression in standard form $a+bi$, where $a$ is the real part and $b$ is the imaginary part.

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). This involves multiplying each part of the first complex number by each part of the second, and then combining like terms, remembering that i² = -1. This process is crucial for finding the product of two complex numbers.

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 multiplying complex numbers, the result should be simplified to this form, which makes it easier to interpret and use in further calculations. Converting to standard form is a key step in ensuring clarity and consistency in complex number operations.