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Question Number 82843 by M±th+et£s last updated on 24/Feb/20

$$showthat$$$$\frac{{(1+\sqrt{3}i)}^4{(1+i)}^8}{{(cos100°-isin100)}^3}=-256$$

Answered by mind is power last updated on 24/Feb/20

$$cos\left(100\right)-isin\left(100\right)=cos\left(\frac{5\pi }{9}\right)-isin\left(\frac{5\pi }{9}\right)$$$$=e^{-\frac{i5\pi }{9}}$$$$\frac{{(1+i\sqrt{3})}^4{(1+i)}^8}{{\left(e^{-\frac{i5\pi }{9}}\right)}^3}=\frac{{\left(2(\frac{1}{2}+\frac{i\sqrt{3}}{2})\right)}^4{\left(\sqrt{2}(\frac{1}{\sqrt{2}}+\frac{i}{\sqrt{2}})\right)}^8}{e^{-\frac{5i\pi }{3}}}$$$$=\frac{2^8{\left(e^{i\frac{\pi }{3}}\right)}^4{\left(e^{\frac{i\pi }{4}}\right)}^8}{e^{-5i\frac{\pi }{3}}}=\frac{2^8{e}^{i(\frac{4\pi }{3}+2\pi +\frac{5\pi }{3})}}{}=2^8{e}^{i\left(5\pi \right)}=2^8(-1)=-256$$

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