Solution 3.1:3b - Revision history

Tek at 10:16, 30 September 2008

2008-09-30T10:16:52Z

←Older revision		Revision as of 10:16, 30 September 2008
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-	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 at 12:52, 22 September 2008

2008-09-22T12:52:39Z

←Older revision		Revision as of 12:52, 22 September 2008
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-	{{~~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

←Older revision

Revision as of 09:10, 9 September 2008

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

2008-08-21T06:52:09Z

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

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2008-04-01T13:53:18Z

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