91Ó°ÊÓ

Given the complex number \(z=\frac{\sqrt{3}}{2}+\frac{1}{2}i\), we first need to determine its modulus and argument. The modulus (magnitude) of \(z\), denoted as \(|z|\), is given by \(|z|=\sqrt{\text{Re}(z)^{2}+\text{Im}(z)^{2}}\). In this case, \(\text{Re}(z)=\frac{\sqrt{3}}{2}\) and \(\text{Im}(z)=\frac{1}{2}\), so we have: $$|z|=\sqrt{\left(\frac{\sqrt{3}}{2}\right)^{2}+\left(\frac{1}{2}\right)^{2}}=\sqrt{\frac{3}{4}+\frac{1}{4}}=\sqrt{1}=1$$ The argument (angle) of \(z\), denoted as \(\text{Arg}(z)\), can be found using trigonometry or by observation, considering the complex number lies in the first quadrant. In this case, $$\text{Arg}(z)=\arctan\left(\frac{\frac{1}{2}}{\frac{\sqrt{3}}{2}}\right)=\arctan\left(\frac{\sqrt{3}}{3}\right)=\frac{\pi}{6}$$ Thus, the polar form of \(z\) is $$z=1\left(\cos{\frac{\pi}{6}} + i\sin{\frac{\pi}{6}}\right)$$

Next, we want to find the value of \(z^{10}\). DeMoivre's theorem states that for a complex number in polar form \(z=r(\cos{\theta}+i\sin{\theta})\), raised to the power of \(n\), where \(n\in\mathbb{Z}\), its result can be found by multiplying the modulus by itself \(n\) times and applying the power to the angle: $$z^n=r^n\left(\cos{(n\theta)}+i\sin{(n\theta)}\right)$$ Applying this to our complex number, which has \(n=10\), we get: $$z^{10}=(1)^{10}\left(\cos{\left(10\frac{\pi}{6}\right)}+i\sin{\left(10\frac{\pi}{6}\right)}\right)$$ Since \((1)^{10}=1\), we have: $$z^{10}=\cos{\frac{5\pi}{3}}+i\sin{\frac{5\pi}{3}}$$

To express the result in the form \(a+bi\), we need to find the real and imaginary parts of \(z^{10}\): $$a=\text{Re}(z^{10})=\cos{\frac{5\pi}{3}}=\frac{1}{2}$$ $$b=\text{Im}(z^{10})=\sin{\frac{5\pi}{3}}=-\frac{\sqrt{3}}{2}$$ Finally, we can write the result in rectangular form: $$z^{10}=\frac{1}{2}-\frac{\sqrt{3}}{2} i$$