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Question

# Now that you have an efficient way to raise a complex number to an integer power, how can you raise a complex number to a fractional power? Start with a simple case, $z^{2}=i$. The goal is to determine the values of z that make the equation a true statement. These values are the two square roots of i, or $i^{1/2}$. a. Write i in polar form. b. Use De Moivre's Theorem to show that $i^{n}$ can be written as $\cos \left(\frac{n \pi}{2}\right)+i \sin \left(\frac{n \pi}{2}\right)$. c. To solve for z, each side of the equation is raised to the $\frac{1}{2}$ power. Use part (b) to write $z=i^{1 / 2}$ in polar form. d. Write $z=i^{1 / 2}$ in rectangular form. Algebraically confirm that $z^{2}=i$. e. The original equation has $z^{2}$, so there are two possible values for z. You determined one solution in part (c). Determine the other solution, then graph both solutions.

Solution

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$\textbf{(a)}$ $i$ in polar form

\begin{aligned} r &=& |i| = \sqrt{(1)^2} = \sqrt{1} \\ r &=& 1 \end{aligned}

The complex number $0+i$ is in Quadrant I.


\begin{aligned} \theta &=& \tan^{-1}\left(\frac{1}{0}\right) \\ \theta &=& \tan^{-1}\left(\infty\right) \\ \theta &=& \frac{\pi}{2} \end{aligned}

Therefore, the polar form of the complex number, $i$ is,


$\left(\cos\left(\frac{\pi}{2}\right)+i\sin\left(\frac{\pi}{2}\right)\right)$.

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