Lösung 3.1:4c

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Each term in the expression can be simplified by breaking down the number under the root sign into its factors,
Each term in the expression can be simplified by breaking down the number under the root sign into its factors,
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{{Displayed math||<math>\begin{align}
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{{Abgesetzte Formel||<math>\begin{align}
50 &= 5\cdot 10 = 5\cdot 5\cdot 2 = 2\cdot 5^{2}\,,\\[5pt]
50 &= 5\cdot 10 = 5\cdot 5\cdot 2 = 2\cdot 5^{2}\,,\\[5pt]
20 &= 2\cdot 10 = 2\cdot 2\cdot 5 = 2^{2}\cdot 5\,,\\[5pt]
20 &= 2\cdot 10 = 2\cdot 2\cdot 5 = 2^{2}\cdot 5\,,\\[5pt]
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and then taking the squares out from under the root sign,
and then taking the squares out from under the root sign,
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{{Displayed math||<math>\begin{align}
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{{Abgesetzte Formel||<math>\begin{align}
\sqrt{50} &= \sqrt{2\cdot 5^2} = 5\sqrt{2}\,,\\
\sqrt{50} &= \sqrt{2\cdot 5^2} = 5\sqrt{2}\,,\\
\sqrt{20} &= \sqrt{2^2\cdot 5} = 2\sqrt{5}\,,\\
\sqrt{20} &= \sqrt{2^2\cdot 5} = 2\sqrt{5}\,,\\
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All together, we get
All together, we get
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{{Displayed math||<math>\begin{align}
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{{Abgesetzte Formel||<math>\begin{align}
\sqrt{50} + 4\sqrt{20} - 3\sqrt{18} - 2\sqrt{80}
\sqrt{50} + 4\sqrt{20} - 3\sqrt{18} - 2\sqrt{80}
&= 5\sqrt{2} + 4\cdot 2\sqrt{5} - 3\cdot 3\sqrt{2} - 2\cdot 4\sqrt{5}\\[5pt]
&= 5\sqrt{2} + 4\cdot 2\sqrt{5} - 3\cdot 3\sqrt{2} - 2\cdot 4\sqrt{5}\\[5pt]

Version vom 08:37, 22. Okt. 2008

Each term in the expression can be simplified by breaking down the number under the root sign into its factors,

\displaystyle \begin{align}

50 &= 5\cdot 10 = 5\cdot 5\cdot 2 = 2\cdot 5^{2}\,,\\[5pt] 20 &= 2\cdot 10 = 2\cdot 2\cdot 5 = 2^{2}\cdot 5\,,\\[5pt] 18 &= 2\cdot 9 = 2\cdot 3\cdot 3 = 2\cdot 3^{2}\,,\\[5pt] 80 &= 8\cdot 10 = (2\cdot 4)\cdot (2\cdot 5) = (2\cdot 2\cdot 2)\cdot (2\cdot 5) = 2^{4}\cdot 5\,, \end{align}

and then taking the squares out from under the root sign,

\displaystyle \begin{align}

\sqrt{50} &= \sqrt{2\cdot 5^2} = 5\sqrt{2}\,,\\ \sqrt{20} &= \sqrt{2^2\cdot 5} = 2\sqrt{5}\,,\\ \sqrt{18} &= \sqrt{2\cdot 3^2} = 3\sqrt{2}\,,\\ \sqrt{80} &= \sqrt{2^4\cdot 5} = 2^{2}\sqrt{5} = 4\sqrt{5}\,\textrm{.} \end{align}

All together, we get

\displaystyle \begin{align}

\sqrt{50} + 4\sqrt{20} - 3\sqrt{18} - 2\sqrt{80} &= 5\sqrt{2} + 4\cdot 2\sqrt{5} - 3\cdot 3\sqrt{2} - 2\cdot 4\sqrt{5}\\[5pt] &= 5\sqrt{2} + 8\sqrt{5} - 9\sqrt{2} - 8\sqrt{5}\\[5pt] &= (5-9)\sqrt{2} + (8-8)\sqrt{5} = -4\sqrt{2}\,\textrm{.} \end{align}