Given the complex number $z=\sqrt{3} e^{\wedge} i \pi / 3-i$.
a) Determine the number $z$ in rectangular form. b) Determine the modulus and main argument of the number $z^{\wedge} 17$

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∻Solution by Steps∻ ‖ step 1 ⋮ Given the complex number $z = \sqrt{3} e^{i \pi / 3} - i$, we need to convert it to rectangular form. ‖ ‖ step 2 ⋮ The term $\sqrt{3} e^{i \pi / 3}$ can be expressed in rectangular form using Euler's formula: $e^{i heta} = \cos( heta) + i \sin( heta)$. ‖ ‖ step 3 ⋮ For $ heta = \pi / 3$, we have $\cos(\pi / 3) = \frac{1}{2}$ and $\sin(\pi / 3) = \frac{\sqrt{3}}{2}$. Thus, $\sqrt{3} e^{i \pi / 3} = \sqrt{3} (\frac{1}{2} + i \frac{\sqrt{3}}{2}) = \frac{\sqrt{3}}{2} + i \frac{3}{2}$. ‖ ‖ step 4 ⋮ Subtracting $i$ from $\frac{\sqrt{3}}{2} + i \frac{3}{2}$, we get $z = \frac{\sqrt{3}}{2} + i \frac{3}{2} - i = \frac{\sqrt{3}}{2} + i (\frac{3}{2} - 1) = \frac{\sqrt{3}}{2} + i \frac{1}{2}$. ‖ ‖ step 5 ⋮ To find the modulus and argument of $z^{17}$, we first find the modulus and argument of $z$. ‖ ‖ step 6 ⋮ The modulus of $z$ is $|z| = \sqrt{(\frac{\sqrt{3}}{2})^2 + (\frac{1}{2})^2} = \sqrt{\frac{3}{4} + \frac{1}{4}} = \sqrt{1} = 1$. ‖ ‖ step 7 ⋮ The argument of $z$ is $\arg(z) = an^{-1}(\frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}}) = an^{-1}(\frac{1}{\sqrt{3}}) = \frac{\pi}{6}$. ‖ ‖ step 8 ⋮ For $z^{17}$, the modulus is $|z^{17}| = |z|^{17} = 1^{17} = 1$. ‖ ‖ step 9 ⋮ The argument of $z^{17}$ is $\arg(z^{17}) = 17 \cdot \arg(z) = 17 \cdot \frac{\pi}{6} = \frac{17 \pi}{6}$. ‖ ‖ step 10 ⋮ Since the argument must be within $[0, 2\pi)$, we reduce $\frac{17 \pi}{6}$ modulo $2\pi$: $\frac{17 \pi}{6} - 2\pi = \frac{17 \pi}{6} - \frac{12 \pi}{6} = \frac{5 \pi}{6}$. ‖ ∻Answer∻ ⚹ The rectangular form of $z$ is $\frac{\sqrt{3}}{2} + i \frac{1}{2}$. The modulus of $z^{17}$ is $1$, and the main argument is $\frac{5 \pi}{6}$. ⚹ ∻Key Concept∻ ⚹Conversion between polar and rectangular forms of complex numbers⚹ ∻Explanation∻ ⚹To convert a complex number from polar to rectangular form, use Euler's formula. The modulus and argument of a complex number raised to a power can be found using properties of exponents and trigonometric identities.⚹◊What is the rectangular form of the complex number $z = \sqrt{3} e^{i \pi / 3} - i$?⍭ Generate me a similar question◊]

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