Question
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 Carrillo John. in the United States
Jan 27,2025
Real Tutor Solution
Tutor-Verified Answer
Answer
The product is \( -36 - 42i \).
Solution
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:**
\[
(8 - 2i)(-3 - 6i)
\]
2. **Use the distributive property (FOIL method):**
\[
= 8 \times (-3) + 8 \times (-6i) + (-2i) \times (-3) + (-2i) \times (-6i)
\]
\[
= -24 - 48i + 6i + 12i^2
\]
3. **Simplify using \( i^2 = -1 \):**
\[
= -24 - 48i + 6i + 12(-1)
\]
\[
= -24 - 48i + 6i - 12
\]
\[
= -36 - 42i
\]
**Final Answer:**
\[
-36 - 42i
\]
Reviewed and approved by the UpStudy tutoring team
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Simplify this solution The Deep Dive
To find the product of \( (8-2i)(-3-6i) \), we use the distributive property (also known as the FOIL method for two binomials): 1. Multiply the first terms: \( 8 \times -3 = -24 \). 2. Multiply the outer terms: \( 8 \times -6i = -48i \). 3. Multiply the inner terms: \( -2i \times -3 = 6i \). 4. Multiply the last terms: \( -2i \times -6i = 12i^2 \) (Recall \( i^2 = -1 \), so \( 12i^2 = -12 \)). Now, combine all these results: \[ -24 - 48i + 6i - 12. \] Combine like terms: \[ (-24 - 12) + (-48i + 6i) = -36 - 42i. \] Thus, \( (8-2i)(-3-6i) = -36 - 42i \). So the result in standard form is: \[ -36 - 42i - \square. \]
