What is the square root of $-16$?
A. $-8i$
B. $-4i$
C. $4i$
D. $8i
Answer (2)
Express − 16 as 16 × ( − 1 ) .
Take the square root: − 16 = 16 × ( − 1 ) .
Separate the terms: 16 × − 1 .
Simplify using i = − 1 : − 16 = 4 i . The answer is 4 i .
Explanation
Understanding the Problem We are asked to find the square root of − 16 . Since the square root of a negative number involves imaginary numbers, we need to remember that the imaginary unit is defined as i = − 1 .
Expressing -16 We can express − 16 as a product of 16 and − 1 : − 16 = 16 × ( − 1 ) .
Taking the Square Root Now, we take the square root of both sides: − 16 = 16 × ( − 1 ) .
Separating the Terms Using the property of square roots, we can separate the terms: 16 × ( − 1 ) = 16 × − 1 .
Substituting Values Since 16 = 4 and − 1 = i , we substitute these values: − 16 = 4 i .
Final Answer Therefore, the square root of − 16 is 4 i .
Examples
Imaginary numbers might seem abstract, but they're incredibly useful in electrical engineering. For example, when analyzing alternating current (AC) circuits, imaginary numbers help represent the phase difference between voltage and current. This allows engineers to calculate impedance, power, and other critical parameters, ensuring efficient and stable circuit designs. Without imaginary numbers, analyzing AC circuits would be much more complex!
The square root of − 16 is 4 i , making option C the correct choice. This result is obtained by separating the square root of the product of 16 and -1, leading to the use of the imaginary unit i . Understanding these properties is vital for mathematical problems involving complex numbers.
;
