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)
Expand the expression ( 4 + 6 i ) 2 using the formula ( a + b ) 2 = a 2 + 2 ab + b 2 .
Simplify the expression, remembering that i 2 = − 1 .
The expanded form is 4 2 + 2 ( 4 ) ( 6 i ) + ( 6 i ) 2 = 16 + 48 i + 36 i 2 .
Since i 2 = − 1 , we have 16 + 48 i − 36 = − 20 + 48 i , so the final answer is − 20 + 48 i .
Explanation
Understanding the Problem We are asked to find an expression equivalent to ( 4 + 6 i ) 2 . This involves expanding the square of a complex number.
Expanding the Expression To find the equivalent expression, we need to expand ( 4 + 6 i ) 2 . We can use the formula ( a + b ) 2 = a 2 + 2 ab + b 2 , where a = 4 and b = 6 i .
Applying the Formula Expanding the expression, we get: ( 4 + 6 i ) 2 = 4 2 + 2 ( 4 ) ( 6 i ) + ( 6 i ) 2
Simplifying the Terms Now, we simplify each term: 4 2 = 16 2 ( 4 ) ( 6 i ) = 48 i ( 6 i ) 2 = 36 i 2
Substituting i^2 = -1 Since i 2 = − 1 , we can substitute this into the expression: 36 i 2 = 36 ( − 1 ) = − 36
Combining Like Terms Now, we combine all the terms: 16 + 48 i − 36 = ( 16 − 36 ) + 48 i = − 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, can be represented as a complex number. Squaring complex numbers is useful when calculating power in AC circuits, where power is related to the square of the current or voltage, both of which can be expressed using complex numbers. This allows engineers to design and optimize electrical systems efficiently.
The expression equivalent to ( 4 + 6 i ) 2 is − 20 + 48 i . This is determined by expanding the binomial and simplifying the terms. The correct answer choice is option A: − 20 + 48 i .
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