Multiply.
[tex]$(-1-2 i)(10+9 i)=$
Answer (1)
Multiply the complex numbers using the distributive property: ( − 1 − 2 i ) ( 10 + 9 i ) = ( − 1 ) ( 10 ) + ( − 1 ) ( 9 i ) + ( − 2 i ) ( 10 ) + ( − 2 i ) ( 9 i ) .
Simplify the expression: − 10 − 9 i − 20 i − 18 i 2 .
Substitute i 2 = − 1 : − 10 − 9 i − 20 i + 18 .
Combine like terms to get the final answer: 8 − 29 i .
Explanation
Understanding the Problem We are asked to multiply two complex numbers: ( − 1 − 2 i ) and ( 10 + 9 i ) .
Applying the Distributive Property We will use the distributive property (also known as the FOIL method) to multiply the two complex numbers:
( − 1 − 2 i ) ( 10 + 9 i ) = ( − 1 ) ( 10 ) + ( − 1 ) ( 9 i ) + ( − 2 i ) ( 10 ) + ( − 2 i ) ( 9 i )
Simplifying the Terms Now, let's simplify each term:
( − 1 ) ( 10 ) = − 10
( − 1 ) ( 9 i ) = − 9 i
( − 2 i ) ( 10 ) = − 20 i
( − 2 i ) ( 9 i ) = − 18 i 2
Combining the Terms So, we have:
− 10 − 9 i − 20 i − 18 i 2
Substituting i 2 = − 1 Recall that i 2 = − 1 . Substitute − 1 for i 2 :
− 10 − 9 i − 20 i − 18 ( − 1 ) = − 10 − 9 i − 20 i + 18
Combining Like Terms Now, combine the real parts and the imaginary parts:
Real part: − 10 + 18 = 8
Imaginary part: − 9 i − 20 i = − 29 i
So, the final result is 8 − 29 i .
Final Answer Therefore, ( − 1 − 2 i ) ( 10 + 9 i ) = 8 − 29 i .
Examples
Complex numbers might seem abstract, but they're incredibly useful in fields like electrical engineering. For example, when analyzing AC circuits, complex numbers help represent the impedance, which is the opposition to the flow of current. Multiplying complex numbers, as we did here, can help determine the overall impedance in a circuit with multiple components. This allows engineers to design efficient and stable electrical systems.
