Simplify each expression.

$-6(3 i)(-2 i)=$

$2(3-i)(-2+4 i)=$

First, multiply the coefficients and imaginary units in the first expression: − 6 ( 3 i ) ( − 2 i ) = 36 i 2 = − 36 .
Next, multiply the complex numbers in the second expression: ( 3 − i ) ( − 2 + 4 i ) = − 6 + 12 i + 2 i − 4 i 2 = − 2 + 14 i .
Multiply the result by 2: 2 ( − 2 + 14 i ) = − 4 + 28 i .
The simplified expressions are − 36 ​ and − 4 + 28 i ​ .

Explanation

Problem Analysis We are given two expressions involving complex numbers and asked to simplify them.

Expression 1: − 6 ( 3 i ) ( − 2 i ) Expression 2: 2 ( 3 − i ) ( − 2 + 4 i )

Simplifying Expression 1 Let's simplify the first expression. We have − 6 ( 3 i ) ( − 2 i ) . First, multiply the coefficients: − 6 × 3 × − 2 = 36 . Then, multiply the imaginary units: i × i = i 2 = − 1 . Therefore, the expression becomes 36 × ( − 1 ) = − 36 .

Simplifying Expression 2 Now, let's simplify the second expression: 2 ( 3 − i ) ( − 2 + 4 i ) . First, we multiply the two complex numbers ( 3 − i ) and ( − 2 + 4 i ) using the distributive property (also known as the FOIL method):


( 3 − i ) ( − 2 + 4 i ) = 3 ( − 2 ) + 3 ( 4 i ) − i ( − 2 ) − i ( 4 i ) = − 6 + 12 i + 2 i − 4 i 2
Since i 2 = − 1 , we have:
− 6 + 12 i + 2 i − 4 ( − 1 ) = − 6 + 12 i + 2 i + 4 = − 2 + 14 i
Now, multiply the result by 2:
2 ( − 2 + 14 i ) = − 4 + 28 i

Final Answer Therefore, the simplified forms of the expressions are:

Expression 1: − 36 Expression 2: − 4 + 28 i
Examples
Complex numbers might seem abstract, but they're incredibly useful in fields like electrical engineering. Imagine designing a circuit where you need to analyze alternating current (AC). AC voltage and current can be represented as complex numbers, making calculations much easier. For example, the impedance of a circuit, which is the opposition to the flow of current, can be expressed as a complex number. By multiplying and dividing these complex impedances, engineers can determine how the circuit will behave, ensuring efficient and stable performance.

Answered by GinnyAnswer | 2025-07-03

The simplifications yield − 36 for the first expression and − 4 + 28 i for the second expression. Multiplying complex numbers involves both coefficient multiplication and using the property of i 2 = − 1 . The final results are clearly presented for both expressions.
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Answered by Anonymous | 2025-07-04