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

Verified step by step guidance

1. Identify the expression to be expanded: $-3i(7i-5)$.
2. Apply the distributive property: multiply $-3i$ by each term inside the parentheses.
3. Calculate $-3i\times 7i$: use the property $i^2=-1$ to simplify.
4. Calculate $-3i\times (-5)$: simplify the multiplication.
5. Combine the results from the previous steps to write the expression in standard form $a+bi$.

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 'i', such as addition, subtraction, multiplication, and division.

Multiplication of Complex Numbers

When multiplying complex numbers, you apply the distributive property, similar to multiplying polynomials. This involves multiplying each term in the first complex number by each term in the second, and then simplifying the result, particularly by substituting 'i^2' with -1. This process is crucial for finding the product of complex expressions.

Standard Form of Complex Numbers

The standard form of a complex number is expressed as a + bi, where 'a' and 'b' are real numbers. In this form, 'a' represents the real part and 'b' represents the imaginary part. Writing complex numbers in standard form is important for clarity and consistency in mathematical communication, especially when performing further operations or comparisons.