Lösung 3.1:2e

Aus Online Mathematik Brückenkurs 1

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Looking first at
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Looking first at <math>\sqrt{18}</math> this square root expression can be simplified by writing 18 as a product of its smallest possible integer factors
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<math>\sqrt{18}</math>
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this square root expression can be simplified by writing
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<math>\text{18}</math>
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as a product of its smallest possible integer factors
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<math>18=2\centerdot 9=2\centerdot 3\centerdot 3=2\centerdot 3^{2}</math>
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{{Displayed math||<math>18 = 2\cdot 9 = 2\cdot 3\cdot 3 = 2\cdot 3^{2}</math>}}
and then we can take the quadratic out of the square root sign by using the rule
and then we can take the quadratic out of the square root sign by using the rule
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<math>\sqrt{a^{2}b}=a\sqrt{b}</math>,
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<math>\sqrt{a^{2}b}=a\sqrt{b}</math> (valid for non-negative ''a'' and ''b''),
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{{Displayed math||<math>\sqrt{18} = \sqrt{2\cdot 3^{2}} = 3\sqrt{2}\,\textrm{.}</math>}}
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<math>\sqrt{18}=\sqrt{2\centerdot 3^{2}}=3\sqrt{2}</math>
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In the same way, we write <math>8 = 2\cdot 4 = 2\cdot 2\cdot 2 = 2^{3}</math> and get
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In the same way, we write
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<math>8=2\centerdot 4=2\centerdot 2\centerdot 2=2^{3}</math>
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and get
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<math>\sqrt{8}=\sqrt{2\centerdot 2^{2}}=2\sqrt{2}</math>
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{{Displayed math||<math>\sqrt{8} = \sqrt{2\cdot 2^{2}} = 2\sqrt{2}\,\textrm{.}</math>}}
All together, we get
All together, we get
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{{Displayed math||<math>\begin{align}
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<math>\begin{align}
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\sqrt{18}\sqrt{8}
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& \sqrt{18}\sqrt{8}=3\sqrt{2}\centerdot 2\sqrt{2}=3\centerdot 2\centerdot \left( \sqrt{2} \right)^{2} \\
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&= 3\sqrt{2}\cdot 2\sqrt{2}\\[5pt]
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& =3\centerdot 2\centerdot 2=12 \\
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&= 3\cdot 2\cdot \bigl(\sqrt{2}\bigr)^{2}\\[5pt]
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& \\
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&= 3\cdot 2\cdot 2\\[5pt]
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\end{align}</math>
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&= 12\,\textrm{.}
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\end{align}</math>}}

Version vom 08:07, 30. Sep. 2008

Looking first at \displaystyle \sqrt{18} this square root expression can be simplified by writing 18 as a product of its smallest possible integer factors

Vorlage:Displayed math

and then we can take the quadratic out of the square root sign by using the rule \displaystyle \sqrt{a^{2}b}=a\sqrt{b} (valid for non-negative a and b),

Vorlage:Displayed math

In the same way, we write \displaystyle 8 = 2\cdot 4 = 2\cdot 2\cdot 2 = 2^{3} and get

Vorlage:Displayed math

All together, we get

Vorlage:Displayed math

Von „http://wiki.sommarmatte.se/wikis/2009/bridgecourse1-TU-Berlin/index.php/L%C3%B6sung_3.1:2e“