Lösung 3.1:1f

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K (Robot: Automated text replacement (-{{Displayed math +{{Abgesetzte Formel))
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Let's begin by calculating some powers of ''i'',
Let's begin by calculating some powers of ''i'',
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{{Displayed math||<math>\begin{align}
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{{Abgesetzte Formel||<math>\begin{align}
i^2 &= i\cdot i = -1\,,\\[5pt]
i^2 &= i\cdot i = -1\,,\\[5pt]
i^3 &= i^2\cdot i = (-1)\cdot i = -i\,,\\[5pt]
i^3 &= i^2\cdot i = (-1)\cdot i = -i\,,\\[5pt]
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Now, we observe that because <math>i^4=1</math>, we can try to factorize <math>i^{11}</math> and <math>i^{20}</math> in terms of <math>i^4</math>,
Now, we observe that because <math>i^4=1</math>, we can try to factorize <math>i^{11}</math> and <math>i^{20}</math> in terms of <math>i^4</math>,
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{{Displayed math||<math>\begin{align}
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{{Abgesetzte Formel||<math>\begin{align}
i^{11} &= i^{4+4+3} = i^4\cdot i^4\cdot i^3 = 1\cdot 1 \cdot (-i) = -i\,,\\[5pt]
i^{11} &= i^{4+4+3} = i^4\cdot i^4\cdot i^3 = 1\cdot 1 \cdot (-i) = -i\,,\\[5pt]
i^{20} &= i^{4+4+4+4+4} = i^4\cdot i^4\cdot i^4\cdot i^4\cdot i^4 = 1\cdot 1 \cdot 1\cdot 1 \cdot 1 = 1\,\textrm{.}
i^{20} &= i^{4+4+4+4+4} = i^4\cdot i^4\cdot i^4\cdot i^4\cdot i^4 = 1\cdot 1 \cdot 1\cdot 1 \cdot 1 = 1\,\textrm{.}
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The answer becomes
The answer becomes
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{{Displayed math||<math>i^{20}+i^{11}=1-i\,\textrm{.}</math>}}
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{{Abgesetzte Formel||<math>i^{20}+i^{11}=1-i\,\textrm{.}</math>}}

Version vom 13:05, 10. Mär. 2009

Let's begin by calculating some powers of i,

\displaystyle \begin{align}

i^2 &= i\cdot i = -1\,,\\[5pt] i^3 &= i^2\cdot i = (-1)\cdot i = -i\,,\\[5pt] i^4 &= i^2\cdot i^2 = (-1)\cdot (-1) = 1\,\textrm{.} \end{align}

Now, we observe that because \displaystyle i^4=1, we can try to factorize \displaystyle i^{11} and \displaystyle i^{20} in terms of \displaystyle i^4,

\displaystyle \begin{align}

i^{11} &= i^{4+4+3} = i^4\cdot i^4\cdot i^3 = 1\cdot 1 \cdot (-i) = -i\,,\\[5pt] i^{20} &= i^{4+4+4+4+4} = i^4\cdot i^4\cdot i^4\cdot i^4\cdot i^4 = 1\cdot 1 \cdot 1\cdot 1 \cdot 1 = 1\,\textrm{.} \end{align}

The answer becomes

\displaystyle i^{20}+i^{11}=1-i\,\textrm{.}
Von „http://wiki.sommarmatte.se/wikis/2009/bridgecourse2-TU-Berlin/index.php/L%C3%B6sung_3.1:1f“