In Exercises 1–10, perform the indicated operations and write the result in standard form.3+4i / 4−2i

Verified step by step guidance

1. insert step 1: Identify the complex numbers involved in the division. The numerator is \(3 + 4i\) and the denominator is \(4 - 2i\).
2. insert step 2: To divide complex numbers, multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of \(4 - 2i\) is \(4 + 2i\).
3. insert step 3: Multiply the numerator \((3 + 4i)\) by the conjugate of the denominator \((4 + 2i)\). Use the distributive property: \((3 + 4i)(4 + 2i)\).
4. insert step 4: Multiply the denominator \((4 - 2i)\) by its conjugate \((4 + 2i)\). This results in a real number: \((4 - 2i)(4 + 2i) = 4^2 - (2i)^2\).
5. insert step 5: Simplify the expression obtained from step 3 and step 4. Write the result in standard form \(a + bi\), where \(a\) and \(b\) are real numbers.

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 (where i² = -1). Understanding complex numbers is essential for performing operations such as addition, subtraction, multiplication, and division.

Division of Complex Numbers

To divide complex numbers, one typically multiplies the numerator and denominator by the conjugate of the denominator. The conjugate of a complex number a + bi is a - bi. This process eliminates the imaginary part in the denominator, allowing for a simplified expression in standard form.

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, the goal is often to express the result in this form, which makes it easier to interpret and use in further calculations. This involves separating the real and imaginary components after performing the necessary arithmetic.