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Geometrische Reihe/k-te Wurzel aus 2/Einheitsnachweis/Aufgabe/Lösung
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Geometrische Reihe/k-te Wurzel aus 2/Einheitsnachweis/Aufgabe
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{\displaystyle {}{\begin{aligned}&\,\,\,\,\,\,\,{\left(1-{\sqrt[{k}]{2}}\right)}\cdot {\left(1+{\sqrt[{k}]{2}}+{\sqrt[{k}]{2}}^{2}+\cdots +{\sqrt[{k}]{2}}^{k-2}+{\sqrt[{k}]{2}}^{k-1}\right)}\\&=1+{\sqrt[{k}]{2}}+{\sqrt[{k}]{2}}^{2}+\cdots +{\sqrt[{k}]{2}}^{k-2}+{\sqrt[{k}]{2}}^{k-1}-{\sqrt[{k}]{2}}{\left(1+{\sqrt[{k}]{2}}+{\sqrt[{k}]{2}}^{2}+\cdots +{\sqrt[{k}]{2}}^{k-2}+{\sqrt[{k}]{2}}^{k-1}\right)}\\&=1+{\sqrt[{k}]{2}}+{\sqrt[{k}]{2}}^{2}+\cdots +{\sqrt[{k}]{2}}^{k-2}+{\sqrt[{k}]{2}}^{k-1}-{\sqrt[{k}]{2}}-{\sqrt[{k}]{2}}^{2}-{\sqrt[{k}]{2}}^{3}-\cdots -{\sqrt[{k}]{2}}^{k-1}-{\sqrt[{k}]{2}}^{k}\\&=1-{\sqrt[{k}]{2}}^{k}\\&=1-2\\&=-1.\,\end{aligned}}}
Zur gelösten Aufgabe
![{\displaystyle {}{\begin{aligned}&\,\,\,\,\,\,\,{\left(1-{\sqrt[{k}]{2}}\right)}\cdot {\left(1+{\sqrt[{k}]{2}}+{\sqrt[{k}]{2}}^{2}+\cdots +{\sqrt[{k}]{2}}^{k-2}+{\sqrt[{k}]{2}}^{k-1}\right)}\\&=1+{\sqrt[{k}]{2}}+{\sqrt[{k}]{2}}^{2}+\cdots +{\sqrt[{k}]{2}}^{k-2}+{\sqrt[{k}]{2}}^{k-1}-{\sqrt[{k}]{2}}{\left(1+{\sqrt[{k}]{2}}+{\sqrt[{k}]{2}}^{2}+\cdots +{\sqrt[{k}]{2}}^{k-2}+{\sqrt[{k}]{2}}^{k-1}\right)}\\&=1+{\sqrt[{k}]{2}}+{\sqrt[{k}]{2}}^{2}+\cdots +{\sqrt[{k}]{2}}^{k-2}+{\sqrt[{k}]{2}}^{k-1}-{\sqrt[{k}]{2}}-{\sqrt[{k}]{2}}^{2}-{\sqrt[{k}]{2}}^{3}-\cdots -{\sqrt[{k}]{2}}^{k-1}-{\sqrt[{k}]{2}}^{k}\\&=1-{\sqrt[{k}]{2}}^{k}\\&=1-2\\&=-1.\,\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/652a40ac77e921df0a67ee2ba4de3302ce8adae1)
