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Chapter 2: Problem 12

Find each product and write the result in standard form. $$(-4-8 i)(3+i)$$

Short Answer

Expert verified
The product of the two complex numbers is \(-4 - 28i\).

Step by step solution

01

Multiplication of Real Parts

First, consider the multiplication of the real parts of both numbers, without losing any imaginary unit i. It results in: \((-4) * 3 = -12\)
02

Multiplication of Real and Imaginary Parts

Now, let's multiply the real part of the first number with the imaginary part of the second, and the imaginary part of the first number and the real part of the second. The resultant multiplication is added to the result of the first step. In this case, we have: \((-4) * i = -4i\) and \(-8 * 3i = -24i\). The sum of these two gives: \(-4i - 24i = -28i\)
03

Multiplication of Imaginary Parts

Multiply the imaginary parts of both complex numbers, remembering that \(i * i = -1\). It results in: \(-(8i) * i = -8* -1 = 8\)
04

Combine all Parts

Finally, add the result from the multiplication of the real parts to the result of the multiplication of the imaginary parts: \(-12 + 8 = -4\), and append the result from the second step, which is the combined result of the mixed multiplications: \(-4 - 28i\). Note that the final form is a standard form of a complex number (a+bi), where a and b are real numbers and i is the imaginary unit.

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