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What is the simplified product of z 1 and z 2 given that z 1 =8+3i and z 2 =1−4i?

A. 29−29i
B. 11+31i
C. −29+29i
D. 11−31i

User Xeed
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Final Answer:

The simplified product of z₁ and z₂, given that z₁ = 8 + 3i and z₂ = 1 − 4i, is 11 + 31i (Option B).

Step-by-step explanation:

To find the product of z₁ and z₂, you multiply the real parts and the imaginary parts separately. Given z₁ = 8 + 3i and z₂ = 1 − 4i, the real parts are 8 and 1, and the imaginary parts are 3i and -4i. Multiplying the real parts gives 8 * 1 = 8, and multiplying the imaginary parts gives (3i) * (-4i) = -12i². Since i² = -1, the product simplifies to 8 - 12(-1), which is 8 + 12 = 20. Therefore, the simplified product is 20, and the final result is 11 + 31i (Option B).

Understanding complex number operations involves applying the rules of arithmetic to both real and imaginary parts. In this case, the multiplication of z₁ and z₂ follows the distributive property, and the simplification considers the definition of the imaginary unit i. The result, 11 + 31i, is expressed in standard form, where the real and imaginary parts are clearly identified.

In conclusion, the simplified product of z₁ = 8 + 3i and z₂ = 1 − 4i is 11 + 31i. This answer is derived through the application of basic algebraic rules and the properties of complex numbers, providing a clear and concise result for the given multiplication.

User Michel Foucault
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