For what value of a does ${ }^2=\left(\frac{1}{2 i}\right)$?
Answer (1)
Simplify the equation to a 2 = − 2 i .
Test the given options by squaring them.
Find that none of the options satisfy the equation.
Conclude that there is no correct answer among the provided choices.
Explanation
Problem Analysis We are given the equation a 2 = 2 i 1 and asked to find the value of a from the given options: − 3 11 , − 3 7 , 3 7 , 3 11 .
Simplifying the Equation First, let's simplify the right side of the equation. We can multiply the numerator and denominator by i to eliminate the imaginary unit from the denominator: 2 i 1 = 2 i 1 ⋅ i i = 2 i 2 i = 2 ( − 1 ) i = − 2 i So we have a 2 = − 2 i .
Testing the Options Now, we need to find the square root of − 2 i . Let's express the complex number − 2 i in the form x + y i , where x = 0 and y = − 2 1 . We are looking for a value a such that a 2 = − 2 i . Let's test the given options by squaring them:
( − 3 11 ) 2 = 9 121 ≈ 13.44 ( − 3 7 ) 2 = 9 49 ≈ 5.44 ( 3 7 ) 2 = 9 49 ≈ 5.44 ( 3 11 ) 2 = 9 121 ≈ 13.44
None of these squared values are equal to − 2 i , which is a complex number. Therefore, none of the given options are correct.
Finding the Actual Value of a To find the actual value of a , we need to solve a 2 = − 2 i . We can write − 2 i in polar form as 2 1 e i ( 3 π /2 + 2 kπ ) , where k is an integer. Then, taking the square root gives us a = ± 2 1 e i ( 3 π /4 + kπ ) = ± 2 1 e i ( 3 π /4 + kπ ) For k = 0 , we have a = ± 2 1 e i ( 3 π /4 ) = ± 2 1 ( cos ( 4 3 π ) + i sin ( 4 3 π ) ) = ± 2 1 ( − 2 1 + i 2 1 ) = ± ( − 2 1 + i 2 1 ) So a = 2 1 − 2 i or a = − 2 1 + 2 i .
Conclusion Since none of the given options match the actual values of a , we can conclude that there is no correct answer among the provided choices.
Examples
Complex numbers are used in electrical engineering to represent alternating currents and voltages. The imaginary unit 'i' helps in modeling the phase difference between voltage and current, which is crucial for designing efficient circuits and analyzing their behavior. For example, the impedance of a circuit, which is the opposition to the flow of alternating current, is often expressed as a complex number.
