Perform each operation. Write answers in standard form. -5i(3-i)²

Verified step by step guidance

1. First, recognize that you need to expand the expression \((3-i)^2\). This involves using the formula for squaring a binomial: \((a-b)^2 = a^2 - 2ab + b^2\).
2. Apply the formula: \((3-i)^2 = 3^2 - 2 \cdot 3 \cdot i + i^2\).
3. Calculate each part: \(3^2 = 9\), \(-2 \cdot 3 \cdot i = -6i\), and \(i^2 = -1\).
4. Combine these results: \(9 - 6i - 1\), which simplifies to \(8 - 6i\).
5. Now, multiply \(-5i\) by the result \(8 - 6i\) using the distributive property: \(-5i(8) + (-5i)(-6i)\).

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.

Imaginary Unit (i)

The imaginary unit 'i' is defined as the square root of -1. It is a fundamental concept in complex number theory, allowing for the extension of the real number system to include solutions to equations that do not have real solutions, such as x² + 1 = 0. Operations involving 'i' require careful handling of its properties, particularly that i² = -1.

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, it is important to simplify the result into this form for clarity and consistency. This involves combining like terms and applying the property that i² = -1 to eliminate any instances of 'i' squared.