Calculate the power and then select your answer from the choices. Make sure your answer is fully simplified. \[ (-2 i)^{5} \]

To calculate \((-2i)^5\), start by using the properties of exponents. First, we recognize that \((-2)^5\) is \(-32\), and \(i^5\) can be simplified using the cycle of powers of \(i\): - \(i^1 = i\) - \(i^2 = -1\) - \(i^3 = -i\) - \(i^4 = 1\) - And then it repeats. Therefore, \(i^5 = i\). Putting it all together: \[ (-2i)^5 = (-2)^5 \cdot i^5 = -32 \cdot i = -32i. \] So, the simplified answer is \(-32i\). Now let’s add some engaging and fun insights about this! Exponentials may feel intimidating, but they follow a delightful rhythm! One of the fun aspects of complex numbers is their captivating circular nature on the complex plane, showcasing how multiplication can rotate and scale numbers in unexpected ways. In the world of mathematics, complex numbers and their powers pop up just about everywhere—from electrical engineering to quantum physics—they help us zap through calculations that would otherwise send our heads spinning. Embrace the magic of \(i\) and watch how you can unlock new dimensions in problem-solving!