Pregunta
Find the product and write the result in standard form \[ (3-4 i)(-5-2 i) \]
Ask by Martinez Elliott. in the United States
Jan 27,2025
Solución de tutoría real
Respuesta verificada por el tutor
Responder
The product is \(-23 + 14i\).
Solución
To find the product \((3 - 4i)(-5 - 2i)\) and express it in standard form \(a + bi\), follow these steps:
1. **Expand the Product:**
\[
(3 - 4i)(-5 - 2i) = 3 \cdot (-5) + 3 \cdot (-2i) + (-4i) \cdot (-5) + (-4i) \cdot (-2i)
\]
2. **Calculate Each Term:**
\[
= -15 - 6i + 20i + 8i^2
\]
3. **Simplify Using \(i^2 = -1\):**
\[
= -15 - 6i + 20i + 8(-1)
\]
\[
= -15 - 6i + 20i - 8
\]
4. **Combine Like Terms:**
\[
(-15 - 8) + (-6i + 20i) = -23 + 14i
\]
**Final Answer:**
\[
-23 + 14i
\]
Revisado y aprobado por el equipo de tutoría de UpStudy
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Simplifique esta solución Beyond the Answer
To find the product of \( (3-4i)(-5-2i) \), we can use the distributive property (also known as the FOIL method for binomials): \[ (3-4i)(-5-2i) = 3 \cdot (-5) + 3 \cdot (-2i) - 4i \cdot (-5) - 4i \cdot (-2i). \] Calculating each term, we have: - \( 3 \cdot (-5) = -15 \) - \( 3 \cdot (-2i) = -6i \) - \( -4i \cdot (-5) = 20i \) - \( -4i \cdot (-2i) = 8i^2 \) (Recall that \( i^2 = -1 \), so this becomes \( 8(-1) = -8 \)) Now, combine all the results: \[ -15 - 6i + 20i - 8 = -15 - 8 + (20i - 6i) = -23 + 14i. \] Thus, the product in standard form is: \[ \boxed{-23 + 14i}. \]
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