Simplify. Write your response in $a+b i$ form.
$(11-8 i)(11+8 i) =$
Answer (2)
Recognize the expression as a difference of squares: ( 11 − 8 i ) ( 11 + 8 i ) .
Apply the difference of squares formula: 1 1 2 − ( 8 i ) 2 = 121 − 64 i 2 .
Substitute i 2 = − 1 : 121 − 64 ( − 1 ) = 121 + 64 .
Simplify to get the final answer: 185 .
Explanation
Understanding the Problem We are asked to simplify the expression ( 11 − 8 i ) ( 11 + 8 i ) and write the result in the form a + bi , where a and b are real numbers. This looks like a product of complex conjugates.
Recognizing the Pattern We can recognize that the given expression is in the form of a difference of squares: ( x − y ) ( x + y ) = x 2 − y 2 , where x = 11 and y = 8 i .
Applying the Formula Applying the difference of squares formula, we have ( 11 − 8 i ) ( 11 + 8 i ) = 1 1 2 − ( 8 i ) 2 .
Simplifying the Expression Now, we simplify the expression. We know that 1 1 2 = 121 and ( 8 i ) 2 = 64 i 2 . So, we have 121 − 64 i 2 .
Using the Definition of i Recall that i 2 = − 1 . Substituting this into the expression, we get 121 − 64 ( − 1 ) = 121 + 64 .
Adding the Numbers Finally, we add the numbers: 121 + 64 = 185 .
Writing the Final Answer We write the result in the form a + bi . In this case, a = 185 and b = 0 , so the simplified expression is 185 + 0 i . Therefore, ( 11 − 8 i ) ( 11 + 8 i ) = 185 + 0 i = 185 .
Examples
Complex numbers might seem abstract, but they're incredibly useful in electrical engineering. Imagine designing a circuit where you need to analyze alternating current (AC). The voltage and current can be represented as complex numbers, and operations like multiplication (as we did here) help engineers calculate power and impedance. Simplifying expressions with complex numbers allows for efficient circuit design and analysis, ensuring that electronic devices work as expected. This ensures stable and reliable performance in devices like smartphones, computers, and power grids.
To simplify ( 11 − 8 i ) ( 11 + 8 i ) , we recognize it as a difference of squares. Calculating leads us to 121 + 64 = 185 . Thus, the final answer is 185 + 0 i .
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