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Now, express the right side in polar form using De Moivre's theorem: $$x^6=(-1)^1 \cdot 3^6[\cos(180^{\circ})+i\sin (180^{\circ})]$$ #Step 2: Take the 6th root of both sides We will take the 6th root of both sides of the equation to solve for x: $$x=\sqrt[6]{(-1)^1 \cdot 3^6[\cos(180^{\circ})+i\sin(180^{\circ})]}$$ Now, we can apply De Moivre's theorem to the right side: $$x=3[\cos\left(\frac{180^{\circ}}{6}\right)+i\sin\left(\frac{180^{\circ}}{6}\right)]$$ #Step 3: Find all solutions for x Since we are solving a degree-6 equation, there should be 6 complex solutions for x. These solutions can be derived by uniformly increasing the angle within the cosine and sine terms by \(360^{\circ}/6 = 60^{\circ}\). The six solutions can be written as: $$x_k=3\left[\cos\left(\frac{180^{\circ}+60^{\circ} k}{6}\right)+i\sin\left(\frac{180^{\circ}+60^{\circ} k}{6}\right)\right], k=0,1,2,3,4,5$$ Short Answer: The six complex solutions for the equation \(x^6+729=0\) are given by: $$x_k=3\left[\cos\left(\frac{180^{\circ}+60^{\circ} k}{6}\right)+i\sin\left(\frac{180^{\circ}+60^{\circ} k}{6}\right)\right], k=0,1,2,3,4,5$$