Simplify. Write your response in $a+b i$ form.
$(4+10 i)^2=$
Answer (2)
Expand the expression: ( 4 + 10 i ) 2 = ( 4 + 10 i ) ( 4 + 10 i ) .
Apply the distributive property: ( 4 + 10 i ) ( 4 + 10 i ) = 16 + 40 i + 40 i + 100 i 2 .
Simplify using i 2 = − 1 : 16 + 80 i − 100 .
Combine real terms to get the final answer: − 84 + 80 i .
Explanation
Understanding the Problem We are asked to simplify ( 4 + 10 i ) 2 and express the result in the form a + bi , where a and b are real numbers. This involves squaring a complex number, which means we need to expand the square and combine the real and imaginary terms.
Expanding the Expression To simplify the expression ( 4 + 10 i ) 2 , we multiply ( 4 + 10 i ) by itself: ( 4 + 10 i ) 2 = ( 4 + 10 i ) ( 4 + 10 i ) We use the distributive property (also known as the FOIL method) to expand this product: ( 4 + 10 i ) ( 4 + 10 i ) = 4 ( 4 ) + 4 ( 10 i ) + 10 i ( 4 ) + 10 i ( 10 i ) = 16 + 40 i + 40 i + 100 i 2
Simplifying the Terms Now, we combine the imaginary terms: 16 + 40 i + 40 i + 100 i 2 = 16 + 80 i + 100 i 2 Recall that i 2 = − 1 . We substitute this into the expression: 16 + 80 i + 100 i 2 = 16 + 80 i + 100 ( − 1 ) = 16 + 80 i − 100
Combining Real Terms Finally, we combine the real terms: 16 + 80 i − 100 = ( 16 − 100 ) + 80 i = − 84 + 80 i So, the simplified expression is − 84 + 80 i .
Final Answer The simplified form of ( 4 + 10 i ) 2 is − 84 + 80 i , which is in the form a + bi , where a = − 84 and b = 80 .
Thus, ( 4 + 10 i ) 2 = − 84 + 80 i .
Examples
Complex numbers might seem abstract, but they're incredibly useful in electrical engineering. For example, when analyzing AC circuits, impedance (a measure of opposition to current) is often expressed as a complex number. Squaring a complex number representing impedance can help calculate power dissipation or voltage drops in the circuit. This allows engineers to design efficient and stable electrical systems.
The simplified form of ( 4 + 10 i ) 2 is − 84 + 80 i , where − 84 is the real part and 80 is the coefficient of the imaginary part. We expanded the square and combined like terms to reach this result.
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