Perform the indicated operations, and write the result in the form $a+b i$. (Simplify your answers completely.)
$(4-3 i)(5+4 i)$
Answer (1)
Multiply the two complex numbers using the distributive property: ( 4 − 3 i ) ( 5 + 4 i ) = 4 ( 5 ) + 4 ( 4 i ) − 3 i ( 5 ) − 3 i ( 4 i ) .
Simplify the expression: 20 + 16 i − 15 i − 12 i 2 .
Substitute i 2 = − 1 : 20 + 16 i − 15 i + 12 .
Combine real and imaginary parts to get the final answer: 32 + i .
Explanation
Understanding the Problem We are asked to multiply two complex numbers, ( 4 − 3 i ) and ( 5 + 4 i ) , and express the result in the standard form a + bi , where a and b are real numbers. The imaginary unit i is defined as i = − 1 , which implies i 2 = − 1 .
Multiplying the Complex Numbers To multiply the two complex numbers, we use the distributive property (also known as the FOIL method). This means we multiply each term in the first complex number by each term in the second complex number: ( 4 − 3 i ) ( 5 + 4 i ) = 4 ( 5 ) + 4 ( 4 i ) − 3 i ( 5 ) − 3 i ( 4 i ) Now, let's perform each of these multiplications.
Expanding the Expression Performing the multiplications, we get: 4 ( 5 ) = 20 4 ( 4 i ) = 16 i − 3 i ( 5 ) = − 15 i − 3 i ( 4 i ) = − 12 i 2 So, our expression becomes: 20 + 16 i − 15 i − 12 i 2
Substituting i 2 = − 1 Now, we recall that i 2 = − 1 . We substitute this into our expression: 20 + 16 i − 15 i − 12 ( − 1 ) 20 + 16 i − 15 i + 12
Combining Like Terms Next, we combine the real parts (20 and 12) and the imaginary parts (16i and -15i): ( 20 + 12 ) + ( 16 i − 15 i ) 32 + 1 i 32 + i
Final Result Therefore, the product of the two complex numbers is 32 + i . This is in the form a + bi , where a = 32 and b = 1 .
Examples
Complex numbers might seem abstract, but they're incredibly useful in fields like electrical engineering. Imagine you're designing an AC circuit. The voltage and current aren't constant; they oscillate. Complex numbers allow engineers to represent these oscillating quantities in a simple algebraic form, making circuit analysis much easier. By using complex impedance, engineers can calculate the behavior of circuits with resistors, capacitors, and inductors. This helps in designing efficient and stable power systems, audio equipment, and communication devices.
