Lösung 3.1:3b - Versionsgeschichte

Moni um 10:10, 9. Aug. 2009

2009-08-09T10:10:25Z

← Nächstältere Version		Version vom 10:10, 9. Aug. 2009
Zeile 13:		Zeile 13:
	{{Abgesetzte Formel\|\|<math>\begin{align}		{{Abgesetzte Formel\|\|<math>\begin{align}
	\sqrt{96} &= \sqrt{2^{5}\cdot 3} = \sqrt{2^{2}\cdot 2^{2}\cdot 2\cdot 3} = 2\cdot 2\cdot \sqrt{2}\cdot \sqrt{3}\,,\\[5pt]		\sqrt{96} &= \sqrt{2^{5}\cdot 3} = \sqrt{2^{2}\cdot 2^{2}\cdot 2\cdot 3} = 2\cdot 2\cdot \sqrt{2}\cdot \sqrt{3}\,,\\[5pt]
-	\sqrt{18} &= \sqrt{2\cdot 3^{2}} = 3\cdot\sqrt{2}\,,	+	\sqrt{18} &= \sqrt{2\cdot 3^{2}} = 3\cdot\sqrt{2}\,.
	\end{align}</math>}}		\end{align}</math>}}

-	Für den Bruch gilt also:	+	Für den Bruch gilt also

	{{Abgesetzte Formel\|\|<math>\frac{\sqrt{96}}{\sqrt{18}} = \frac{2\cdot 2\cdot \sqrt{2}\cdot \sqrt{3}}{3\cdot \sqrt{2}} = \frac{4\sqrt{3}}{3}\,\textrm{.}</math>}}		{{Abgesetzte Formel\|\|<math>\frac{\sqrt{96}}{\sqrt{18}} = \frac{2\cdot 2\cdot \sqrt{2}\cdot \sqrt{3}}{3\cdot \sqrt{2}} = \frac{4\sqrt{3}}{3}\,\textrm{.}</math>}}

Markus.bez um 09:40, 10. Jun. 2009

2009-06-10T09:40:53Z

← Nächstältere Version		Version vom 09:40, 10. Jun. 2009
Zeile 1:		Zeile 1:
	Wir zerlegen alle Zahlen im Ausdruck in ihre Primfaktoren, um den Ausdruck zu vereinfachen.		Wir zerlegen alle Zahlen im Ausdruck in ihre Primfaktoren, um den Ausdruck zu vereinfachen.

-	Durch mehrfache Division mit 2 und 3, erhalten wir	+	Durch mehrfache Division mit 2 und 3 erhalten wir

	{{Abgesetzte Formel\|\|<math>\begin{align}		{{Abgesetzte Formel\|\|<math>\begin{align}
Zeile 9:		Zeile 9:
	\end{align}</math>}}		\end{align}</math>}}

-	~~und also,~~	+	Also

	{{Abgesetzte Formel\|\|<math>\begin{align}		{{Abgesetzte Formel\|\|<math>\begin{align}
Zeile 16:		Zeile 16:
	\end{align}</math>}}		\end{align}</math>}}

-	~~Der~~ Bruch ~~kann~~ also ~~wie~~	+	Für den Bruch gilt also:

	{{Abgesetzte Formel\|\|<math>\frac{\sqrt{96}}{\sqrt{18}} = \frac{2\cdot 2\cdot \sqrt{2}\cdot \sqrt{3}}{3\cdot \sqrt{2}} = \frac{4\sqrt{3}}{3}\,\textrm{.}</math>}}		{{Abgesetzte Formel\|\|<math>\frac{\sqrt{96}}{\sqrt{18}} = \frac{2\cdot 2\cdot \sqrt{2}\cdot \sqrt{3}}{3\cdot \sqrt{2}} = \frac{4\sqrt{3}}{3}\,\textrm{.}</math>}}

Martin um 20:53, 23. Mär. 2009

2009-03-23T20:53:55Z

← Nächstältere Version		Version vom 20:53, 23. Mär. 2009
Zeile 1:		Zeile 1:
-	When simplifying a radical expression, a common technique is to divide up the numbers under the root sign into their smallest possible integer factors and then take out the squares and see if common factors cancel each other out or can be combined together in ~~a new way~~.	+	Wir zerlegen alle Zahlen im Ausdruck in ihre Primfaktoren, um den Ausdruck zu vereinfachen.

-	~~By successively dividing by~~ 2 ~~and~~ 3, ~~we see that~~	+	Durch mehrfache Division mit 2 und 3, erhalten wir

	{{Abgesetzte Formel\|\|<math>\begin{align}		{{Abgesetzte Formel\|\|<math>\begin{align}
Zeile 9:		Zeile 9:
	\end{align}</math>}}		\end{align}</math>}}

-	~~Thus~~,	+	und also,

	{{Abgesetzte Formel\|\|<math>\begin{align}		{{Abgesetzte Formel\|\|<math>\begin{align}
Zeile 16:		Zeile 16:
	\end{align}</math>}}		\end{align}</math>}}

-	~~and the whole quotient can be written as~~	+	Der Bruch kann also wie

	{{Abgesetzte Formel\|\|<math>\frac{\sqrt{96}}{\sqrt{18}} = \frac{2\cdot 2\cdot \sqrt{2}\cdot \sqrt{3}}{3\cdot \sqrt{2}} = \frac{4\sqrt{3}}{3}\,\textrm{.}</math>}}		{{Abgesetzte Formel\|\|<math>\frac{\sqrt{96}}{\sqrt{18}} = \frac{2\cdot 2\cdot \sqrt{2}\cdot \sqrt{3}}{3\cdot \sqrt{2}} = \frac{4\sqrt{3}}{3}\,\textrm{.}</math>}}

-		+	Hinweis: Wir können auch mit Potenzen rechnen
-	~~Note~~: ~~If it is difficult to work with radicals, it is possible instead to write everything in power form~~	+

	{{Abgesetzte Formel\|\|<math>\begin{align}		{{Abgesetzte Formel\|\|<math>\begin{align}
Zeile 33:		Zeile 32:
	= \frac{4\sqrt{3}}{3}\,\textrm{.}		= \frac{4\sqrt{3}}{3}\,\textrm{.}
	\end{align}</math>}}		\end{align}</math>}}
-
-	(In the last equality, we multiply top and bottom by <math>\sqrt{3}</math>.)

Tekbot: hat „Solution 3.1:3b“ nach „Lösung 3.1:3b“ verschoben: Robot: moved page

2008-10-22T14:20:25Z

hat „Solution 3.1:3b“ nach „Lösung 3.1:3b“ verschoben: Robot: moved page

← Nächstältere Version

Version vom 14:20, 22. Okt. 2008

Tekbot: Robot: Automated text replacement (-{{Displayed math +{{Abgesetzte Formel)

2008-10-22T08:36:58Z

Robot: Automated text replacement (-{{Displayed math +{{Abgesetzte Formel)

← Nächstältere Version		Version vom 08:36, 22. Okt. 2008
Zeile 3:		Zeile 3:
	By successively dividing by 2 and 3, we see that		By successively dividing by 2 and 3, we see that

-	{{~~Displayed math~~\|\|<math>\begin{align}	+	{{Abgesetzte Formel\|\|<math>\begin{align}
	96 &= 2\cdot 48 = 2\cdot 2\cdot 24 = 2\cdot 2\cdot 2\cdot 12 = 2\cdot 2\cdot 2\cdot 2\cdot 6\\		96 &= 2\cdot 48 = 2\cdot 2\cdot 24 = 2\cdot 2\cdot 2\cdot 12 = 2\cdot 2\cdot 2\cdot 2\cdot 6\\
	&= 2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 3 = 2^{5}\cdot 3,\\[5pt]		&= 2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 3 = 2^{5}\cdot 3,\\[5pt]
Zeile 11:		Zeile 11:
	Thus,		Thus,

-	{{~~Displayed math~~\|\|<math>\begin{align}	+	{{Abgesetzte Formel\|\|<math>\begin{align}
	\sqrt{96} &= \sqrt{2^{5}\cdot 3} = \sqrt{2^{2}\cdot 2^{2}\cdot 2\cdot 3} = 2\cdot 2\cdot \sqrt{2}\cdot \sqrt{3}\,,\\[5pt]		\sqrt{96} &= \sqrt{2^{5}\cdot 3} = \sqrt{2^{2}\cdot 2^{2}\cdot 2\cdot 3} = 2\cdot 2\cdot \sqrt{2}\cdot \sqrt{3}\,,\\[5pt]
	\sqrt{18} &= \sqrt{2\cdot 3^{2}} = 3\cdot\sqrt{2}\,,		\sqrt{18} &= \sqrt{2\cdot 3^{2}} = 3\cdot\sqrt{2}\,,
Zeile 18:		Zeile 18:
	and the whole quotient can be written as		and the whole quotient can be written as

-	{{~~Displayed math~~\|\|<math>\frac{\sqrt{96}}{\sqrt{18}} = \frac{2\cdot 2\cdot \sqrt{2}\cdot \sqrt{3}}{3\cdot \sqrt{2}} = \frac{4\sqrt{3}}{3}\,\textrm{.}</math>}}	+	{{Abgesetzte Formel\|\|<math>\frac{\sqrt{96}}{\sqrt{18}} = \frac{2\cdot 2\cdot \sqrt{2}\cdot \sqrt{3}}{3\cdot \sqrt{2}} = \frac{4\sqrt{3}}{3}\,\textrm{.}</math>}}


	Note: If it is difficult to work with radicals, it is possible instead to write everything in power form		Note: If it is difficult to work with radicals, it is possible instead to write everything in power form

-	{{~~Displayed math~~\|\|<math>\begin{align}	+	{{Abgesetzte Formel\|\|<math>\begin{align}
	\frac{\sqrt{96}}{\sqrt{18}}		\frac{\sqrt{96}}{\sqrt{18}}
	&= \frac{96^{1/2}}{18^{1/2}}		&= \frac{96^{1/2}}{18^{1/2}}

Tek um 10:16, 30. Sep. 2008

2008-09-30T10:16:52Z

← Nächstältere Version		Version vom 10:16, 30. Sep. 2008
Zeile 1:		Zeile 1:
-	When simplifying a ~~root~~ expression, a common technique is to divide up the numbers under the root sign into their smallest possible integer factors and then take out the squares and see if common ~~factor~~ cancel each other out or can be combined together in a new way.	+	When simplifying a radical expression, a common technique is to divide up the numbers under the root sign into their smallest possible integer factors and then take out the squares and see if common factors cancel each other out or can be combined together in a new way.

-	By successively dividing by	+	By successively dividing by 2 and 3, we see that
-	~~<math>~~2~~</math>~~	+
-	and	+
-	~~<math>~~3~~</math>~~, we see that	+
-		+
-		+
-	~~<math>\begin{align}~~	+
-	~~& 96=2\centerdot 48=2\centerdot 2\centerdot 24=2\centerdot 2\centerdot 2\centerdot 12=2\centerdot 2\centerdot 2\centerdot 2\centerdot 6 \\~~	+
-	~~& =2\centerdot 2\centerdot 2\centerdot 2\centerdot 2\centerdot 3=2^{5}\centerdot 3, \\~~	+
-	~~& 18=2\centerdot 9=2\centerdot 3\centerdot 3=2\centerdot 3^{2}. \\~~	+
-	~~\end{align}</math>~~	+

		+	{{Displayed math\|\|<math>\begin{align}
		+	96 &= 2\cdot 48 = 2\cdot 2\cdot 24 = 2\cdot 2\cdot 2\cdot 12 = 2\cdot 2\cdot 2\cdot 2\cdot 6\\
		+	&= 2\cdot 2\cdot 2\cdot 2\cdot 2\cdot 3 = 2^{5}\cdot 3,\\[5pt]
		+	18 &= 2\cdot 9 = 2\cdot 3\cdot 3 = 2\cdot 3^{2}.
		+	\end{align}</math>}}

	Thus,		Thus,

-		+	{{Displayed math\|\|<math>\begin{align}
-	<math>\begin{align}	+	\sqrt{96} &= \sqrt{2^{5}\cdot 3} = \sqrt{2^{2}\cdot 2^{2}\cdot 2\cdot 3} = 2\cdot 2\cdot \sqrt{2}\cdot \sqrt{3}\,,\\[5pt]
-	& \sqrt{96}=\sqrt{2^{5}\~~centerdot~~ 3}=\sqrt{2^{2}\~~centerdot~~ 2^{2}\~~centerdot~~ 2\~~centerdot~~ 3}=2\~~centerdot~~ 2\~~centerdot~~ \sqrt{2}\~~centerdot~~ \sqrt{3} \\	+	\sqrt{18} &= \sqrt{2\cdot 3^{2}} = 3\cdot\sqrt{2}\,,
-	& \sqrt{18}=\sqrt{2\~~centerdot~~ 3^{2}}=3\~~centerdot~~ \sqrt{2} \\	+	\end{align}</math>}}
-	\end{align}</math>	+
-		+

	and the whole quotient can be written as		and the whole quotient can be written as

-		+	{{Displayed math\|\|<math>\frac{\sqrt{96}}{\sqrt{18}} = \frac{2\cdot 2\cdot \sqrt{2}\cdot \sqrt{3}}{3\cdot \sqrt{2}} = \frac{4\sqrt{3}}{3}\,\textrm{.}</math>}}
-	<math>\frac{\sqrt{96}}{\sqrt{18}}=\frac{2\~~centerdot~~ 2\~~centerdot~~ \sqrt{2}\~~centerdot~~ \sqrt{3}}{3\~~centerdot~~ \sqrt{2}}=\frac{4\sqrt{3}}{3}</math>	+
-		+
-		+
-	~~NOTE: If it is difficult to work with the root sign, it is possible instead to write everything in power form~~	+


-	~~<math>\begin{align}~~	+	Note: If it is difficult to work with radicals, it is possible instead to write everything in power form
-	& \frac{\sqrt{96}}{\sqrt{18}}=\frac{\left( 96 \right)^{\frac{1}{2}}}{\left( 18 \right)^{{1}/{2}\;}}=\frac{\left( 2^{5}\centerdot 3 \right)^{\frac{1}{2}}}{\left( 2\centerdot 3^{2} \right)^{{1}/{2}\;}}=\frac{2^{5\centerdot \frac{1}{2}}\centerdot 3^{\frac{1}{2}}}{2^{\frac{1}{2}}\centerdot 3^{2\centerdot \frac{1}{2}}} \\	+
-	~~& =2^{\frac{5}{2}-\frac{1}{2}}\centerdot 3^{\frac{1}{2}-1}=2^{2}\centerdot 3^{-\frac{1}{2}}=\frac{4}{\sqrt{3}}=\frac{4\sqrt{3}}{3} \\~~	+
-	~~\end{align}</math>~~	+

		+	{{Displayed math\|\|<math>\begin{align}
		+	\frac{\sqrt{96}}{\sqrt{18}}
		+	&= \frac{96^{1/2}}{18^{1/2}}
		+	= \frac{(2^{5}\cdot 3)^{1/2}}{(2\cdot 3^{2})^{1/2}}
		+	= \frac{2^{5\cdot\frac{1}{2}}\cdot 3^{\frac{1}{2}}}{2^{\frac{1}{2}}\cdot 3^{2\cdot \frac{1}{2}}}\\[5pt]
		+	&= 2^{\frac{5}{2}-\frac{1}{2}}\cdot 3^{\frac{1}{2}-1}
		+	= 2^{2}\cdot 3^{-\frac{1}{2}}
		+	= \frac{4}{\sqrt{3}}
		+	= \frac{4\sqrt{3}}{3}\,\textrm{.}
		+	\end{align}</math>}}

-	(in the last ~~line~~, we multiply top and bottom by	+	(In the last equality, we multiply top and bottom by <math>\sqrt{3}</math>.)
-	<math>\sqrt{3}</math>	+
-	).	+

Ian um 12:52, 22. Sep. 2008

2008-09-22T12:52:39Z

← Nächstältere Version		Version vom 12:52, 22. Sep. 2008
Zeile 1:		Zeile 1:
-	{{~~NAVCONTENT_START}~~}	+	When simplifying a root expression, a common technique is to divide up the numbers under the root sign into their smallest possible integer factors and then take out the squares and see if common factor cancel each other out or can be combined together in a new way.
-	~~<center> [[Image:3_1_3b-1(~~2).~~gif]]~~ </~~center~~>	+
-	{{~~NAVCONTENT_STOP~~}}	+	By successively dividing by
-	{{~~NAVCONTENT_START~~}}	+	<math>2</math>
-	<~~center~~> ~~[[Image~~:~~3_1_3b-~~2(2)~~.gif]] <~~/~~center>~~	+	and
-	{{~~NAVCONTENT_STOP~~}}	+	<math>3</math>, we see that
		+
		+
		+	<math>\begin{align}
		+	& 96=2\centerdot 48=2\centerdot 2\centerdot 24=2\centerdot 2\centerdot 2\centerdot 12=2\centerdot 2\centerdot 2\centerdot 2\centerdot 6 \\
		+	& =2\centerdot 2\centerdot 2\centerdot 2\centerdot 2\centerdot 3=2^{5}\centerdot 3, \\
		+	& 18=2\centerdot 9=2\centerdot 3\centerdot 3=2\centerdot 3^{2}. \\
		+	\end{align}</math>
		+
		+
		+	Thus,
		+
		+
		+	<math>\begin{align}
		+	& \sqrt{96}=\sqrt{2^{5}\centerdot 3}=\sqrt{2^{2}\centerdot 2^{2}\centerdot 2\centerdot 3}=2\centerdot 2\centerdot \sqrt{2}\centerdot \sqrt{3} \\
		+	& \sqrt{18}=\sqrt{2\centerdot 3^{2}}=3\centerdot \sqrt{2} \\
		+	\end{align}</math>
		+
		+
		+	and the whole quotient can be written as
		+
		+
		+	<math>\frac{\sqrt{96}}{\sqrt{18}}=\frac{2\centerdot 2\centerdot \sqrt{2}\centerdot \sqrt{3}}{3\centerdot \sqrt{2}}=\frac{4\sqrt{3}}{3}</math>
		+
		+
		+	NOTE: If it is difficult to work with the root sign, it is possible instead to write everything in power form
		+
		+
		+	<math>\begin{align}
		+	& \frac{\sqrt{96}}{\sqrt{18}}=\frac{\left( 96 \right)^{\frac{1}{2}}}{\left( 18 \right)^{{1}/{2}\;}}=\frac{\left( 2^{5}\centerdot 3 \right)^{\frac{1}{2}}}{\left( 2\centerdot 3^{2} \right)^{{1}/{2}\;}}=\frac{2^{5\centerdot \frac{1}{2}}\centerdot 3^{\frac{1}{2}}}{2^{\frac{1}{2}}\centerdot 3^{2\centerdot \frac{1}{2}}} \\
		+	& =2^{\frac{5}{2}-\frac{1}{2}}\centerdot 3^{\frac{1}{2}-1}=2^{2}\centerdot 3^{-\frac{1}{2}}=\frac{4}{\sqrt{3}}=\frac{4\sqrt{3}}{3} \\
		+	\end{align}</math>
		+
		+
		+	(in the last line, we multiply top and bottom by
		+	<math>\sqrt{3}</math>
		+	).

Tekbot: Lösning 3.1:3b moved to Solution 3.1:3b: Robot: moved page

2008-09-09T09:10:03Z

Lösning 3.1:3b moved to Solution 3.1:3b: Robot: moved page

← Nächstältere Version

Version vom 09:10, 9. Sep. 2008

Tekbot: Robot: Automated text replacement (-[[Bild: +[[Image:)

2008-08-21T06:52:09Z

Robot: Automated text replacement (-[[Bild: +[[Image:)

← Nächstältere Version		Version vom 06:52, 21. Aug. 2008
Zeile 1:		Zeile 1:
	{{NAVCONTENT_START}}		{{NAVCONTENT_START}}
-	<center> [[~~Bild~~:3_1_3b-1(2).gif]] </center>	+	<center> [[Image:3_1_3b-1(2).gif]] </center>
	{{NAVCONTENT_STOP}}		{{NAVCONTENT_STOP}}
	{{NAVCONTENT_START}}		{{NAVCONTENT_START}}
-	<center> [[~~Bild~~:3_1_3b-2(2).gif]] </center>	+	<center> [[Image:3_1_3b-2(2).gif]] </center>
	{{NAVCONTENT_STOP}}		{{NAVCONTENT_STOP}}

Oskar: Ny sida: {{NAVCONTENT_START}} Bild:3_1_3b-1(2).gif {{NAVCONTENT_STOP}} {{NAVCONTENT_START}} Bild:3_1_3b-2(2).gif {{NAVCONTENT_STOP}}

2008-04-01T13:53:18Z

Ny sida: {{NAVCONTENT_START}} <center> Bild:3_1_3b-1(2).gif </center> {{NAVCONTENT_STOP}} {{NAVCONTENT_START}} <center> Bild:3_1_3b-2(2).gif </center> {{NAVCONTENT_STOP}}

Neue Seite

{{NAVCONTENT_START}}
<center> [[Bild:3_1_3b-1(2).gif]] </center>
{{NAVCONTENT_STOP}}
{{NAVCONTENT_START}}
<center> [[Bild:3_1_3b-2(2).gif]] </center>
{{NAVCONTENT_STOP}}