Pregunta
\( 1 \leftarrow \quad \begin{array}{l}\text { Multiply } \\ (6+3 i)(9+3 i)\end{array} \)
Ask by Craig Tyler. in the United States
Feb 28,2025
Solución de inteligencia artificial de Upstudy
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The product of (6 + 3i) and (9 + 3i) is 45 + 45i.
Solución
To multiply (6 + 3i)(9 + 3i), use the distributive property (or FOIL):
1. Multiply the first terms: 6 × 9 = 54
2. Multiply the outer terms: 6 × 3i = 18i
3. Multiply the inner terms: 3i × 9 = 27i
4. Multiply the last terms: 3i × 3i = 9i²
Now combine the results:
54 + 18i + 27i + 9i²
= 54 + 45i + 9i²
Recall that i² = -1, so substitute:
9i² = 9(–1) = –9
Thus:
54 + 45i – 9 = 45 + 45i
So, the product is 45 + 45i.
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Simplifique esta solución Bonus Knowledge
To multiply the complex numbers \( (6 + 3i)(9 + 3i) \), you can use the distributive property (also known as the FOIL method for binomials). First, calculate the real and imaginary parts separately: 1. **Real Part:** \( 6 \cdot 9 = 54 \) 2. **Outer:** \( 6 \cdot 3i = 18i \) 3. **Inner:** \( 3i \cdot 9 = 27i \) 4. **Imaginary Part:** \( 3i \cdot 3i = 9i^2 \) (remember that \( i^2 = -1 \), so this becomes \(-9\)) Now combine these results: - Real part: \( 54 - 9 = 45 \) - Imaginary part: \( 18i + 27i = 45i \) Therefore, the product is \( 45 + 45i \). Now you can impress your friends with your complex multiplication skills!
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