Perform the indicated operations and write the result in standard form. 3+4i / 4−2i

Verified step by step guidance

1. Identify the given complex division problem: \(\frac{3+4i}{4-2i}\).
2. To simplify the division of complex numbers, multiply both the numerator and the denominator by the conjugate of the denominator. The conjugate of \$4-2i\( is \)4+2i$.
3. Multiply the numerator: \((3+4i)(4+2i)\) using the distributive property (FOIL method).
4. Multiply the denominator: \((4-2i)(4+2i)\), which is a difference of squares and simplifies to \$4^2 - (2i)^2$.
5. After performing the multiplications, separate the real and imaginary parts in the numerator and divide each by the real number obtained in the denominator to write the result in standard form \(a + bi\).

Key Concepts

Complex Numbers and Standard Form

Complex numbers are expressed as a + bi, where a is the real part and b is the imaginary part. The standard form refers to writing the result explicitly in this form, separating real and imaginary components for clarity.

Division of Complex Numbers

Dividing complex numbers involves multiplying numerator and denominator by the conjugate of the denominator to eliminate the imaginary part in the denominator. This process simplifies the expression into standard form.

Complex Conjugate

The complex conjugate of a number a + bi is a - bi. Multiplying by the conjugate removes the imaginary part in the denominator, as (a + bi)(a - bi) equals a² + b², a real number, facilitating division.