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Math  /  Algebra

QuestionA. (3+2i)(4+3i)(3+2 i)(4+3 i)

Studdy Solution

STEP 1

1. We are asked to find the product of two complex numbers.
2. The complex numbers are in the form a+bi a + bi , where i i is the imaginary unit with the property i2=−1 i^2 = -1 .

STEP 2

1. Use the distributive property to expand the product.
2. Simplify the expression by combining like terms.

STEP 3

Use the distributive property (also known as the FOIL method for binomials) to expand the expression (3+2i)(4+3i)(3+2i)(4+3i):
(3+2i)(4+3i)=3⋅4+3⋅3i+2i⋅4+2i⋅3i (3+2i)(4+3i) = 3 \cdot 4 + 3 \cdot 3i + 2i \cdot 4 + 2i \cdot 3i

STEP 4

Calculate each term in the expanded expression:
3⋅4=12 3 \cdot 4 = 12 3⋅3i=9i 3 \cdot 3i = 9i 2i⋅4=8i 2i \cdot 4 = 8i 2i⋅3i=6i2 2i \cdot 3i = 6i^2

STEP 5

Combine the real parts and the imaginary parts, and simplify using i2=−1 i^2 = -1 :
12+9i+8i+6i2=12+17i+6(−1) 12 + 9i + 8i + 6i^2 = 12 + 17i + 6(-1)
=12+17i−6 = 12 + 17i - 6

STEP 6

Combine the real numbers:
12−6=6 12 - 6 = 6
Thus, the expression simplifies to:
6+17i 6 + 17i
The product of the complex numbers is:
6+17i \boxed{6 + 17i}

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