Divide and express the result in standard form. 8i/(4 - 3i)

Guida verificata passo dopo passo

Identify the problem: You need to divide the complex number \$8i$ by the complex number $(4 - 3i)$ and express the result in standard form, which is $a + bi$ where $a$ and $b$ are real numbers.

To divide complex numbers, multiply the numerator and denominator by the conjugate of the denominator. The conjugate of $(4 - 3i)$ is $(4 + 3i)$. So multiply both numerator and denominator by $(4 + 3i)$:

\[\frac{8i}{4 - 3i} \times \frac{4 + 3i}{4 + 3i}\]

Use the distributive property (FOIL) to expand both the numerator and the denominator:

- Numerator: $8i \times (4 + 3i)$

- Denominator: $(4 - 3i)(4 + 3i)$

Simplify the denominator using the difference of squares formula: $(a - bi)(a + bi) = a^2 + b^2$. Here, $a=4$ and $b=3$, so the denominator becomes $4^2 + 3^2$.

Risposta video verificata per un problema simile:

Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.

Durata del video:

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

Complex Number Standard Form

The standard form of a complex number is expressed as a + bi, where a and b are real numbers, and i is the imaginary unit with i² = -1. Writing complex numbers in this form separates the real and imaginary parts clearly.

Video consigliato:

05:02

Multiplying Complex Numbers

Division of Complex Numbers

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

Video consigliato:

04:22

Dividing Complex Numbers

Complex Conjugate

The complex conjugate of a number a + bi is a - bi. Multiplying a complex number by its conjugate results in a real number, which helps in rationalizing denominators when dividing complex numbers.

Video consigliato: