Given \\(A=3+4i\\) and \\(B=2+5i\\), find AB
Hi, Cleopas. In order to multiply these two complex numbers, use FOIL (First Outside, Inside Last). This technique gives us AB = (3+4i)(2+5i) = 15i + 6 - 20 + 8i = 23i - 14.
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To find the product \\(AB\\), we can multiply the real and imaginary parts separately and then combine them.
Given:
\\(A = 3 + 4i\\)
\\(B = 2 + 5i\\)
Let's proceed step by step:
Step 1: Multiply the real parts of \\(A\\) and \\(B\\):
\\(3 \times 2 = 6\\)
Step 2: Multiply the imaginary parts of \\(A\\) and \\(B\\):
\\(4i \times 5i = -20\\) (since \\(i^2 = -1\\))
Step 3: Combine the real and imaginary parts:
\\(6 - 20i\\)
Therefore, the product of \\(A\\) and \\(B\\) is \\(AB = 6 - 20i\\).
To find the product of two complex numbers, we can use the distributive property and simplify the expression. Let's find the product of A and B step by step:
Step 1: Start by multiplying the real parts of A and B:
Real part of A = 3
Real part of B = 2
Product of real parts = 3 * 2 = 6
Step 2: Multiply the real part of A with the imaginary part of B and vice versa:
Real part of A = 3
Imaginary part of B = 5i
Product of real part of A and imaginary part of B = 3 * 5i = 15i
Imaginary part of A = 4i
Real part of B = 2
Product of imaginary part of A and real part of B = 4i * 2 = 8i
Step 3: Multiply the imaginary parts of A and B:
Imaginary part of A = 4i
Imaginary part of B = 5i
Product of imaginary parts = 4i * 5i = 20i^2
Step 4: Simplify the product of the imaginary parts using the property i^2 = -1:
20i^2 = 20 * (-1) = -20
Step 5: Combine the results from Steps 1-4 to get the final product AB:
AB = (6 + 15i + 8i - 20)
= (6 + 23i - 20)
= -14 + 23i
Therefore, the product of A and B is -14 + 23i.