Homework: 3.1 Question 2, of 100 points Homework 3.1.VQ-2 Points: 0 of 4 Save uestion list Question 1 Question 2 Question 3 Watch the video and then solve the problem given below. Click here to watch the videe. Express in terms of \( i \). \[ \sqrt{-72} \] \[ \sqrt{-72}=\square \] (Simplify your answer. Type your answer in the form \( a+b i \).)

To express \( \sqrt{-72} \) in terms of \( i \), we start by rewriting it as follows: \[ \sqrt{-72} = \sqrt{72} \cdot \sqrt{-1} \] Since \( \sqrt{-1} = i \), we can simplify \( \sqrt{72} \). Breaking it down gives us \( \sqrt{72} = \sqrt{36 \times 2} = \sqrt{36} \cdot \sqrt{2} = 6\sqrt{2} \). Now, combining everything, we find: \[ \sqrt{-72} = 6\sqrt{2} \cdot i \] So, the final answer in the form \( a + bi \) is: \[ 0 + 6\sqrt{2} i \] Knowing the concept of imaginary numbers comes from the necessity to solve equations where no real number solutions exist, like \( x^2 + 1 = 0 \). Thus, mathematicians introduced \( i \) to extend the number system. Imaginary numbers have practical applications, especially in engineering fields such as electrical engineering. They are used in analyzing AC circuits and understanding waveforms, making them indispensable in today's tech-driven world!