Question
True or False If $z=r e^{i \theta}$ is a complex number and $n$ is an integer, then $z^{n}=r^{n} e^{i \theta}$
Step 1
A complex number $z$ can be written in the form $z = r e^{i \theta}$, where $r$ is the magnitude of $z$ and $\theta$ is the argument of $z$. Show more…
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Key Concepts
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Prerequisites
Complex Numbers
Determine whether the statement is true or false. Let $z_{1}=r_{1}\left(\cos \theta_{1}+i \sin \theta_{1}\right)=r_{1} e^{j \theta_{1}}$ and $z_{2}=r_{2}\left(\cos \theta_{2}+i \sin \theta_{2}\right)=r_{2} e^{i \theta,}$ be two complex numbers. Use the properties of exponentials to show that $\frac{z_{1}}{z_{2}}=\frac{r_{1}}{r_{2}}\left[\cos \left(\theta_{1}-\theta_{2}\right)+i \sin \left(\theta_{1}-\theta_{2}\right)\right]$.
Vectors, the Complex Plane, and Polar Coordinates
Products, Quotients, Powers, and Roots of Complex Numbers
Determine whether the statement is true or false. Let $z_{1}=r_{1}\left(\cos \theta_{1}+i \sin \theta_{1}\right)=r_{1} e^{i \theta_{i}}$ and $z_{2}=r_{2}\left(\cos \theta_{2}+i \sin \theta_{2}\right)=r_{2} e^{i \theta_{2}}$ be two complex numbers. Use the properties of exponentials to show that $z_{1} z_{2}=r_{1} r_{2}\left[\cos \left(\theta_{1}+\theta_{2}\right)+i \sin \left(\theta_{1}+\theta_{2}\right)\right]$
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