Mária mala nájsť algebraický tvar komplexného čísla $(1-\mathrm{i})^8$.
V ktorom kroku riešenia urobila Mária chybu?
Máriino riešenie: $$ \begin{aligned} (1-\mathrm{i})^8 &\stackrel{(1)}= \left[\sqrt2\left(\cos\frac{7\pi}{4}+\mathrm{i}\sin\frac{7\pi}{4}\right)\right]^8 = \cr &\stackrel{(2)}= (\sqrt2)^8\left(8\cdot\cos\frac{7\pi}{4}+8\cdot\mathrm{i}\sin\frac{7\pi}{4}\right) =\cr &\stackrel{(3)}= 2^4(4\sqrt{2}-4\sqrt{2}\mathrm{i}) =\cr&\stackrel{(4)}= 64\sqrt{2}-64\sqrt{2}\mathrm{i} \end{aligned} $$
V kroku (1). Goniometrický tvar čísla $1-\mathrm{i}$ je $\sqrt2\left(\cos\frac{3\pi}{4}+ \mathrm{i}\sin\frac{3\pi}{4}\right)$ a teda: $$(1-\mathrm{i})^8=\left[\sqrt2\left(\cos\frac{3\pi}{4}+\mathrm{i}\sin\frac{3\pi}{4}\right)\right]^8$$
V kroku (2). Správne zjednodušenie je: $$ (\sqrt2)^8\left(\cos\left(8\cdot\frac{7\pi}{4}\right)+\mathrm{i}\sin\left(8\cdot\frac{7\pi}{4}\right)\right) $$
V kroku (3). Správne zjednodušenie je: $$ 2^8(\sqrt{2}-\sqrt{2}\mathrm{i}) $$
V kroku (4). Správne zjednodušenie je:
$$ 128\sqrt{2}-128\sqrt{2}\mathrm{i} $$
$$ \begin{aligned} (1-\mathrm{i})^8 &\stackrel{(1)}= \left[\sqrt2\left(\cos\frac{7\pi}{4}+\mathrm{i}\sin\frac{7\pi}{4}\right)\right]^8 = \cr &\stackrel{(2)}= (\sqrt2)^8\left(\cos\left(8\cdot\frac{7\pi}{4}\right)+\mathrm{i}\sin\left(8\cdot\frac{7\pi}{4}\right)\right) =\cr &\stackrel{(3)}= 2^4(\cos14\pi+\mathrm{i}\sin14\pi) =\cr &\stackrel{(4)}= 16(\cos0+\mathrm{i}\sin0) =\cr&\stackrel{(5)}= 16 \end{aligned} $$