Simplify. Write your response in $a+b i$ form.
$(-6-11 i)(8-7 i)=$
Answer (1)
Expand the product using the distributive property: ( − 6 − 11 i ) ( 8 − 7 i ) = − 48 + 42 i − 88 i + 77 i 2 .
Substitute i 2 = − 1 : − 48 + 42 i − 88 i − 77 .
Combine the real parts: − 48 − 77 = − 125 .
Combine the imaginary parts: 42 i − 88 i = − 46 i . The final answer is − 125 − 46 i .
Explanation
Understanding the Problem We are asked to simplify the expression ( − 6 − 11 i ) ( 8 − 7 i ) and write the result in the form a + bi , where a and b are real numbers.
Expanding the Product To simplify the expression, we will use the distributive property (also known as the FOIL method) to expand the product: ( − 6 − 11 i ) ( 8 − 7 i ) = − 6 ( 8 ) + ( − 6 ) ( − 7 i ) + ( − 11 i ) ( 8 ) + ( − 11 i ) ( − 7 i )
Simplifying Each Term Now, we multiply each term: − 6 ( 8 ) = − 48 ( − 6 ) ( − 7 i ) = 42 i ( − 11 i ) ( 8 ) = − 88 i ( − 11 i ) ( − 7 i ) = 77 i 2 So the expression becomes: − 48 + 42 i − 88 i + 77 i 2
Substituting i 2 = − 1 Recall that i 2 = − 1 . Substitute this into the expression: − 48 + 42 i − 88 i + 77 ( − 1 ) = − 48 + 42 i − 88 i − 77
Combining Like Terms Now, combine the real parts and the imaginary parts: Real parts: − 48 − 77 = − 125 Imaginary parts: 42 i − 88 i = − 46 i So the simplified expression is: − 125 − 46 i
Final Answer Therefore, the simplified form of ( − 6 − 11 i ) ( 8 − 7 i ) is − 125 − 46 i .
Examples
Complex numbers are used in electrical engineering to analyze alternating current (AC) circuits. The impedance of a circuit, which includes resistance and reactance, is represented as a complex number. By performing operations with complex numbers, engineers can calculate the voltage and current in AC circuits, ensuring efficient and safe designs. For example, multiplying complex impedances helps determine the total impedance in a series circuit, which is crucial for designing stable and reliable electrical systems.
