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

Verified step by step guidance

1. Identify the expression to be squared: $(-2+\sqrt{-4}{)}^2$.
2. Recognize that $\sqrt{-4}$ can be rewritten using imaginary numbers: $\sqrt{-4}=\sqrt{4}\cdot \sqrt{-1}=2i$.
3. Substitute $2i$ for $\sqrt{-4}$ in the expression: $(-2+2i{)}^2$.
4. Apply the formula for squaring a binomial: $(a+b{)}^2=a^2+2ab+b^2$, where $a=-2$ and $b=2i$.
5. Calculate each term: $(-2{)}^2=4$, $2(-2)(2i)=-8i$, and $(2i{)}^2=4i^2=4(-1)=-4$.

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. In this problem, √-4 can be rewritten as 2i, allowing us to work with complex numbers.

Standard Form of Complex Numbers

The standard form of a complex number is a + bi, where 'a' and 'b' are real numbers. To express a complex number in standard form, it is essential to separate the real and imaginary components. In the given expression, after performing the operations, the result should be simplified to fit this format.

Binomial Expansion

Binomial expansion refers to the process of expanding expressions that are raised to a power, such as (a + b)^n. The expansion can be done using the binomial theorem, which provides a formula for calculating the coefficients of the terms in the expansion. In this case, we will apply the binomial expansion to (-2 + 2i)^2 to find the result.