Multiply. $(-4+4 i)(-1+5 i)$
Answer (1)
Multiply the complex numbers using the distributive property: ( − 4 + 4 i ) ( − 1 + 5 i ) = ( − 4 ) ( − 1 ) + ( − 4 ) ( 5 i ) + ( 4 i ) ( − 1 ) + ( 4 i ) ( 5 i ) .
Simplify the expression: 4 − 20 i − 4 i + 20 i 2 .
Substitute i 2 = − 1 : 4 − 20 i − 4 i − 20 .
Combine real and imaginary parts to get the final answer: − 16 − 24 i .
Explanation
Understanding the Problem We are asked to multiply two complex numbers: ( − 4 + 4 i ) and ( − 1 + 5 i ) . The product will also be a complex number of the form a + bi , where a and b are real numbers.
Applying the Distributive Property We will multiply the two complex numbers using the distributive property (also known as the FOIL method): ( − 4 + 4 i ) ( − 1 + 5 i ) = ( − 4 ) ( − 1 ) + ( − 4 ) ( 5 i ) + ( 4 i ) ( − 1 ) + ( 4 i ) ( 5 i )
Simplifying the Expression Now, let's simplify each term: ( − 4 ) ( − 1 ) = 4 ( − 4 ) ( 5 i ) = − 20 i ( 4 i ) ( − 1 ) = − 4 i ( 4 i ) ( 5 i ) = 20 i 2 So we have: 4 − 20 i − 4 i + 20 i 2
Substituting i 2 = − 1 Recall that i 2 = − 1 . Substitute this into the expression: 4 − 20 i − 4 i + 20 ( − 1 ) = 4 − 20 i − 4 i − 20
Combining Real and Imaginary Parts Combine the real parts ( 4 e wl in e − 20 ) and the imaginary parts ( − 20 i − 4 i ) :
( 4 − 20 ) + ( − 20 i − 4 i ) = − 16 − 24 i
Final Answer The final result of the multiplication is − 16 − 24 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 (resistance to alternating current) of components like resistors, capacitors, and inductors. By multiplying complex numbers representing impedances, engineers can calculate the total impedance in a circuit, which is crucial for designing efficient and stable electronic devices. This ensures that devices like smartphones and computers receive the correct power and operate reliably.
