Which expression is equivalent to $(4+6 i)^2 ?$
A. $-20+48 i$
B. $8+12 i$
C. $16-36 i$
D. $20+48 i$
Answer (2)
The expression equivalent to ( 4 + 6 i ) 2 is − 20 + 48 i , obtained by expanding the expression and simplifying. The answer corresponds to choice A. − 20 + 48 i .
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Expand the expression ( 4 + 6 i ) 2 to ( 4 + 6 i ) ( 4 + 6 i ) .
Apply the distributive property: 16 + 24 i + 24 i + 36 i 2 .
Combine like terms: 16 + 48 i + 36 i 2 .
Substitute i 2 = − 1 and simplify: − 20 + 48 i . The final answer is − 20 + 48 i .
Explanation
Understanding the problem We are asked to find an expression equivalent to ( 4 + 6 i ) 2 , where i is the imaginary unit. The imaginary unit i is defined as i 2 = − 1 .
Expanding the expression To find the equivalent expression, we need to expand ( 4 + 6 i ) 2 . This means we need to multiply ( 4 + 6 i ) by itself: ( 4 + 6 i ) ( 4 + 6 i ) .
Applying the distributive property We use the distributive property (also known as the FOIL method) to expand the expression:
( 4 + 6 i ) ( 4 + 6 i ) = 4 ( 4 ) + 4 ( 6 i ) + 6 i ( 4 ) + 6 i ( 6 i )
Simplifying terms Now we simplify each term:
4 ( 4 ) = 16 4 ( 6 i ) = 24 i 6 i ( 4 ) = 24 i 6 i ( 6 i ) = 36 i 2
Combining the terms So the expression becomes:
16 + 24 i + 24 i + 36 i 2
Combining like terms Combine the like terms (the terms with i ):
24 i + 24 i = 48 i
So the expression is now:
16 + 48 i + 36 i 2
Substituting i^2 = -1 Since i 2 = − 1 , we substitute − 1 for i 2 :
16 + 48 i + 36 ( − 1 ) = 16 + 48 i − 36
Simplifying to the final answer Now combine the constant terms:
16 − 36 = − 20
So the final expression is:
− 20 + 48 i
Final Answer Therefore, the expression equivalent to ( 4 + 6 i ) 2 is − 20 + 48 i .
Examples
Complex numbers are used in electrical engineering to analyze alternating current circuits. The impedance of a circuit, which is the opposition to the flow of current, is represented as a complex number. For example, if you have a circuit with a resistor and an inductor, the total impedance can be calculated using complex numbers. Squaring complex numbers, as we did in this problem, can be part of calculating power or other important circuit characteristics. Understanding complex number operations is crucial for designing and analyzing such circuits effectively.
