Simplify the expression $(8+6 i)(8-6 i)$.
Answer (2)
The expression ( 8 + 6 i ) ( 8 − 6 i ) simplifies to 100 by using the formula for products of complex conjugates, which is a 2 + b 2 where a = 8 and b = 6 . This yields the calculation 64 + 36 = 100 . Therefore, the final answer is 100 .
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Recognize the expression as a product of complex conjugates.
Apply the formula ( a + bi ) ( a − bi ) = a 2 + b 2 .
Substitute a = 8 and b = 6 into the formula.
Calculate the result: 100 .
Explanation
Understanding the Problem We are asked to simplify the expression ( 8 + 6 i ) ( 8 − 6 i ) . This involves multiplying two complex numbers. Notice that the two complex numbers are complex conjugates of each other.
Using the Conjugate Property Recall that when multiplying complex conjugates, we have ( a + bi ) ( a − bi ) = a 2 + b 2 . In our case, a = 8 and b = 6 .
Substitution Substitute a = 8 and b = 6 into the formula a 2 + b 2 to get 8 2 + 6 2 .
Calculating the Squares Calculate 8 2 = 64 and 6 2 = 36 .
Final Calculation Add the results: 64 + 36 = 100 . Therefore, ( 8 + 6 i ) ( 8 − 6 i ) = 100 .
Examples
Complex numbers might seem abstract, but they're incredibly useful in fields like electrical engineering. For example, when analyzing alternating current (AC) circuits, complex numbers help represent the impedance, which is the opposition to the flow of current. The real part of the impedance represents resistance, while the imaginary part represents reactance (due to capacitors and inductors). By using complex numbers, engineers can easily calculate the total impedance and analyze the behavior of AC circuits. Simplifying expressions like the one in this problem is a fundamental skill in this context.
