In Exercises 1–10, perform the indicated operations and write the result in standard form. __4 + √−8 / 2

Verified step by step guidance

1. Recognize that the expression involves a complex number because of the square root of a negative number.
2. Rewrite the square root of the negative number using the imaginary unit: \( \sqrt{-8} = \sqrt{8} \cdot i \).
3. Simplify \( \sqrt{8} \) to \( 2\sqrt{2} \), so \( \sqrt{-8} = 2\sqrt{2}i \).
4. Substitute \( \sqrt{-8} \) in the original expression: \( \frac{4 + 2\sqrt{2}i}{2} \).
5. Divide each term in the numerator by 2: \( \frac{4}{2} + \frac{2\sqrt{2}i}{2} \).

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 the expression √−8.

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 with complex numbers, it is important to express the result in this form to clearly identify the real and imaginary components. This helps in further calculations and understanding the geometric representation of complex numbers on the complex plane.

Operations with Complex Numbers

Operations with complex numbers include addition, subtraction, multiplication, and division. Each operation follows specific rules, such as combining like terms for addition or using the conjugate for division. Mastery of these operations is crucial for simplifying expressions and solving equations that involve complex numbers, ensuring accurate results in standard form.