In Exercises 37–52, perform the indicated operations and write the result in standard form. __ (−2 + √−4)²

Verified step by step guidance

1. Recognize that the expression involves a complex number: \(-2 + \sqrt{-4}\). Since \(\sqrt{-4} = 2i\), rewrite the expression as \((-2 + 2i)^2\).
2. Recall the formula for squaring a binomial: \((a + b)^2 = a^2 + 2ab + b^2\). Here, \(a = -2\) and \(b = 2i\).
3. Calculate each term separately: \(a^2 = (-2)^2\), \(2ab = 2 \times (-2) \times 2i\), and \(b^2 = (2i)^2\).
4. Simplify each term: \((-2)^2 = 4\), \(2 \times (-2) \times 2i = -8i\), and \((2i)^2 = 4i^2\). Remember that \(i^2 = -1\), so \(4i^2 = 4 \times (-1) = -4\).
5. Combine all terms: \$4 - 8i - 4\(. Then, simplify the real parts \)4 - 4\( and write the expression in standard form \)a + bi$.

Key Concepts

Complex Numbers and Imaginary Unit

Complex numbers consist of a real part and an imaginary part, expressed as a + bi, where i is the imaginary unit with the property i² = -1. Understanding how to interpret and manipulate expressions involving √-4 requires recognizing that √-4 = 2i.

Operations with Complex Numbers

Performing operations like addition, multiplication, and exponentiation on complex numbers follows algebraic rules, treating i as a variable but applying i² = -1 to simplify. Squaring a complex number involves expanding the binomial and simplifying using these rules.

Standard Form of a Complex Number

The standard form of a complex number is a + bi, where a and b are real numbers. After performing operations, the result should be simplified and expressed clearly in this form, separating the real and imaginary parts.