In Exercises 37–52, perform the indicated operations and write the result in standard form. (- 3 - √-7)²

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1. Rewrite the expression (-3 - √-7)^2 using the property of squaring a binomial: (a - b)^2 = a^2 - 2ab + b^2. Here, a = -3 and b = √-7.
2. Square the first term, a^2: (-3)^2 = 9.
3. Multiply the first term and the second term, then double it: -2(-3)(√-7). Simplify this to 6√-7.
4. Square the second term, b^2: (√-7)^2. Recall that √-7 can be rewritten as √7 * i (where i is the imaginary unit, i^2 = -1). Therefore, (√-7)^2 = -7.
5. Combine all the terms: a^2 - 2ab + b^2 = 9 - 6√-7 - 7. Simplify further to write the result in standard form, which separates the real and imaginary parts.

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 square roots of negative numbers, as seen in this problem.

Standard Form of Complex Numbers

The standard form of a complex number is a + bi, where 'a' and 'b' are real numbers. When performing operations on complex numbers, it is important to express the final result in this form to clearly identify the real and imaginary components. This helps in further calculations and interpretations in complex number theory.

Squaring a Binomial

Squaring a binomial involves applying the formula (a + b)² = a² + 2ab + b². In this case, the expression (-3 - √-7) is a binomial that needs to be squared. Recognizing how to expand this expression correctly is crucial for obtaining the correct result, especially when dealing with complex numbers.