Solution 3.1:2g - Revision history

Tek at 10:35, 30 September 2008

2008-09-30T10:35:29Z

←Older revision		Revision as of 10:35, 30 September 2008
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-	~~Because~~	+	Since <math>-125</math> can be written as <math>-125 = (-5)\cdot (-5)\cdot (-5) = (-5)^3</math>, the number <math>\sqrt[3]{-125}</math> is defined as
-	<math>-~~\text{~~125 }</math>	+
-	can be written as	+
-	<math>-125=~~\left~~( -5 ~~\right~~)\~~centerdot \left~~( -5 ~~\right~~)\~~centerdot \left~~( -5 ~~\right~~)=~~\left~~( -5 ~~\right~~)^{3}</math>,	+
-	<math>\sqrt[3]{-125}</math>	+
-	is defined as	+

		+	{{Displayed math\|\|<math>\sqrt[3]{-125}=-5\,\textrm{.}</math>}}

-	<math>\sqrt[3]{-125}=-5</math>

		+	Note: As opposed to <math>\sqrt{-125}</math> (the square root of -125) which is not defined, <math>\sqrt[3]{-125}</math> is defined. In other words, there does not exist any number ''x'' which satisfies <math>x^2 = -125</math>, but there is a number ''x'' which satisfies <math>x^3 = -125\,</math>.

-	NOTE: As opposed to
-	<math>\sqrt{-125}</math>
-	(the square root of
-	<math>-125</math>
-	) which is not defined,
-	<math>\sqrt[3]{-125}</math>
-	is defined . In other words, there does not exist any number which satisfies
-	<math>x^{\text{2}}=-\text{125}</math>, but there is a number
-	<math>x</math>
-	which satisfies
-	<math>x^{\text{3}}=-\text{125}</math>.

-	~~NOTE~~: It is possible to write the calculation in the solution as	+	Note: It is possible to write the calculation in the solution as
-	<math>\sqrt[3]{-125}=\sqrt[3]{\left( -5 \right)^{3}}=\left( -5 \right)^{1}=-5</math>, but one has to be careful when one calculates using negative numbers and fractional exponents. Sometimes, the expression is not defined and the usual power rules do not always hold. Look, for example, at the calculation	+

		+	{{Displayed math\|\|<math>\sqrt[3]{-125} = \sqrt[3]{(-5)^{3}} = (-5)^1 = -5\,,</math>}}

-	<math>\begin{align}	+	but one has to be careful when one calculates using negative numbers and fractional exponents. Sometimes, the expression is not defined and the usual power rules do not always hold. Look, for example, at the calculation
-	& -5=~~\left~~( -125 ~~\right~~)^{{1}/{3~~}\;~~}=~~\left~~( -125 ~~\right~~)^{{2}/{6~~}\;~~}=\~~left~~( ~~\left~~( -125 ~~\right~~)^{2} \~~right~~)^{{1}/{6}~~\;} \\~~	+
-	& =15625^{{1}/{6~~}\;~~}=5 \\	+	{{Displayed math\|\|<math>\begin{align}
-	\end{align}</math>	+	-5 = (-125)^{1/3} = (-125)^{2/6} = \bigl((-125)^2\bigr)^{1/6} = 15625^{1/6}=5\,\textrm{.}
		+	\end{align}</math>}}

Ian at 09:00, 30 September 2008

2008-09-30T09:00:53Z

←Older revision		Revision as of 09:00, 30 September 2008
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-	{{~~NAVCONTENT_START~~}}	+	Because
-	<~~center~~> [~~[Image~~:~~3_1_2g~~.~~gif~~]] </~~center~~>	+	<math>-\text{125 }</math>
-	{{~~NAVCONTENT_STOP~~}}	+	can be written as
		+	<math>-125=\left( -5 \right)\centerdot \left( -5 \right)\centerdot \left( -5 \right)=\left( -5 \right)^{3}</math>,
		+	<math>\sqrt[3]{-125}</math>
		+	is defined as
		+
		+
		+	<math>\sqrt[3]{-125}=-5</math>
		+
		+
		+	NOTE: As opposed to
		+	<math>\sqrt{-125}</math>
		+	(the square root of
		+	<math>-125</math>
		+	) which is not defined,
		+	<math>\sqrt[3]{-125}</math>
		+	is defined . In other words, there does not exist any number which satisfies
		+	<math>x^{\text{2}}=-\text{125}</math>, but there is a number
		+	<math>x</math>
		+	which satisfies
		+	<math>x^{\text{3}}=-\text{125}</math>.
		+
		+	NOTE: It is possible to write the calculation in the solution as
		+	<math>\sqrt[3]{-125}=\sqrt[3]{\left( -5 \right)^{3}}=\left( -5 \right)^{1}=-5</math>, but one has to be careful when one calculates using negative numbers and fractional exponents. Sometimes, the expression is not defined and the usual power rules do not always hold. Look, for example, at the calculation
		+
		+
		+	<math>\begin{align}
		+	& -5=\left( -125 \right)^{{1}/{3}\;}=\left( -125 \right)^{{2}/{6}\;}=\left( \left( -125 \right)^{2} \right)^{{1}/{6}\;} \\
		+	& =15625^{{1}/{6}\;}=5 \\
		+	\end{align}</math>

Tekbot: Lösning 3.1:2g moved to Solution 3.1:2g: Robot: moved page

2008-09-09T09:09:23Z

Lösning 3.1:2g moved to Solution 3.1:2g: Robot: moved page

←Older revision

Revision as of 09:09, 9 September 2008

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

2008-08-21T06:51:49Z

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

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2008-04-01T13:51:26Z

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