Question 20 (5 points)
Multiply the complex numbers: $(1 / 2+4 i)^2$
A) $-15 \frac{3}{4}+4 i$
B) $15^3 / 4+8 i$
C) $15^3 / 4+4 i$
D) $-15^3 / 4+8 i$
Answer (1)
Expand the expression ( 2 1 + 4 i ) 2 using the binomial formula.
Simplify each term: ( 2 1 ) 2 = 4 1 , 2 ( 2 1 ) ( 4 i ) = 4 i , and ( 4 i ) 2 = − 16 .
Combine the real parts: 4 1 − 16 = − 4 63 = − 15 4 3 .
The final result is − 15 4 3 + 4 i .
Explanation
Understanding the Problem We are asked to multiply the complex number ( 2 1 + 4 i ) by itself, which means we need to calculate ( 2 1 + 4 i ) 2 .
Applying the Binomial Formula To solve this, we will use the formula for squaring a binomial: ( a + b ) 2 = a 2 + 2 ab + b 2 . In our case, a = 2 1 and b = 4 i .
Expanding the Expression Now, let's expand the expression: ( 2 1 + 4 i ) 2 = ( 2 1 ) 2 + 2 ( 2 1 ) ( 4 i ) + ( 4 i ) 2
Simplifying Each Term Next, we simplify each term: ( 2 1 ) 2 = 4 1 2 ( 2 1 ) ( 4 i ) = 4 i ( 4 i ) 2 = 16 i 2 Since i 2 = − 1 , we have 16 i 2 = 16 ( − 1 ) = − 16 .
Combining Real and Imaginary Parts Now, we combine the real and imaginary parts: 4 1 + 4 i − 16 To combine the real parts, we need to find a common denominator: 4 1 − 16 = 4 1 − 4 64 = − 4 63 So, the expression becomes: − 4 63 + 4 i
Final Result Converting the improper fraction to a mixed number, we have: − 4 63 = − 15 4 3 Therefore, the final result is: − 15 4 3 + 4 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 often represented as a complex number. Squaring a complex number can be useful in calculating power dissipation or other circuit characteristics. For example, if the voltage across a component is represented by the complex number V and the current through it is represented by the complex number I , the power dissipated can be related to the square of the magnitude of these complex numbers.
