Power with De Moivre

Problem

Using De Moivre, compute $(1+i)^6$.

Solution

Write $1+i$ in exponential form: $\rho=\sqrt 2$, $\theta=\dfrac{\pi}{4}$, hence $1+i=\sqrt 2\,e^{\,i\pi/4}$. By De Moivre’s formula: $(1+i)^6 = (\sqrt 2)^6\,e^{\,i\cdot 6\pi/4} = 8\,e^{\,i\,3\pi/2} = 8\left(\cos\tfrac{3\pi}{2} + i\sin\tfrac{3\pi}{2}\right) = 8(0 - i) = \boxed{-8i}.$

Links

Topics: Complex numbers
Concepts: De Moivre · Exponential form
Methods: De Moivre
Skills: Calculating · Using formulae
People: Abraham de Moivre
Exercise type: Evaluating expressions