Pregunta
Find the product and write the result in standard form. \( (8-2 i)(-3-6 i) \) \( (8-2 i)(-3-6 i)=\square \) (Simplify your answer. Type your answer in the form \( a+b i \)
Ask by Wright Schofield. in the United States
Jan 27,2025
Solución de tutoría real
Respuesta verificada por el tutor
Responder
\[
(8 - 2i)(-3 - 6i) = -36 - 42i
\]
Solución
To find the product of the complex numbers \( (8 - 2i) \) and \( (-3 - 6i) \) and express it in the standard form \( a + bi \), follow these steps:
1. **Multiply the Two Complex Numbers:**
\[
(8 - 2i)(-3 - 6i)
\]
Apply the distributive property (FOIL method):
\[
= 8 \times (-3) + 8 \times (-6i) + (-2i) \times (-3) + (-2i) \times (-6i)
\]
\[
= -24 - 48i + 6i + 12i^2
\]
2. **Simplify the Expression:**
Recall that \( i^2 = -1 \):
\[
= -24 - 48i + 6i + 12(-1)
\]
\[
= -24 - 48i + 6i - 12
\]
3. **Combine Like Terms:**
- **Real Parts:** \( -24 - 12 = -36 \)
- **Imaginary Parts:** \( -48i + 6i = -42i \)
So, the expression simplifies to:
\[
-36 - 42i
\]
**Final Answer:**
\[
(8 - 2i)(-3 - 6i) = -36 - 42i
\]
Revisado y aprobado por el equipo de tutoría de UpStudy
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Simplifique esta solución Bonus Knowledge
To find the product \( (8-2i)(-3-6i) \), we can use the distributive property (also known as the FOIL method for binomials). First, let's distribute: 1. \( 8 \cdot (-3) = -24 \) 2. \( 8 \cdot (-6i) = -48i \) 3. \( -2i \cdot (-3) = 6i \) 4. \( -2i \cdot (-6i) = 12i^2 \) Now we combine these results: \[ -24 - 48i + 6i + 12i^2 \] Recall that \( i^2 = -1 \). Therefore, \( 12i^2 = 12(-1) = -12 \). Now substitute that back in: \[ -24 - 48i + 6i - 12 = -36 - 42i \] Thus, the product simplifies to: \[ (8-2i)(-3-6i) = -36 - 42i \] So, the final result in the form \( a + bi \) is: \[ -36 - 42i \]
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