Multiply. $(5-6 i)(4-4 i)$ Write your answer as a complex number in standard form.
Answer (2)
Multiply the complex numbers using the distributive property: ( 5 − 6 i ) ( 4 − 4 i ) = 20 − 20 i − 24 i + 24 i 2 .
Substitute i 2 = − 1 : 20 − 20 i − 24 i − 24 .
Combine the real and imaginary parts: ( 20 − 24 ) + ( − 20 − 24 ) i .
Simplify to get the final answer: − 4 − 44 i .
Explanation
Understanding the Problem We are asked to multiply two complex numbers, ( 5 − 6 i ) and ( 4 − 4 i ) , and express the result in the standard form a + bi , where a and b are real numbers.
Applying the Distributive Property To multiply the complex numbers, we use the distributive property (also known as the FOIL method): ( 5 − 6 i ) ( 4 − 4 i ) = 5 ( 4 ) + 5 ( − 4 i ) − 6 i ( 4 ) − 6 i ( − 4 i )
Performing the Multiplication Now, we perform each multiplication: 5 ( 4 ) = 20 5 ( − 4 i ) = − 20 i − 6 i ( 4 ) = − 24 i − 6 i ( − 4 i ) = 24 i 2 So we have: 20 − 20 i − 24 i + 24 i 2
Substituting i 2 = − 1 Recall that i 2 = − 1 . Substitute this into the expression: 20 − 20 i − 24 i + 24 ( − 1 ) 20 − 20 i − 24 i − 24
Combining Like Terms Combine the real parts and the imaginary parts: ( 20 − 24 ) + ( − 20 i − 24 i ) − 4 + ( − 44 i ) − 4 − 44 i
Final Answer The product of the two complex numbers in standard form is − 4 − 44 i .
Examples
Complex numbers might seem abstract, but they're incredibly useful in fields like electrical engineering. Imagine you're designing a circuit and need to analyze alternating current (AC). AC voltage and current can be represented as complex numbers, where the real part represents the resistance and the imaginary part represents the reactance (opposition to current change). By multiplying complex numbers, you can calculate the total impedance in the circuit, which helps you predict how the circuit will behave. This ensures your design is efficient and stable, preventing overloads or malfunctions. So, understanding complex number multiplication is crucial for any electrical engineer to analyze and design AC circuits effectively.
To multiply the complex numbers (5-6i) and (4-4i), apply the distributive property. After performing the multiplication, combining like terms, and substituting i 2 = − 1 , we find the result to be − 4 − 44 i .
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