Lösung 3.1:2g

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Because
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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
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<math>-\text{125 }</math>
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can be written as
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<math>-125=\left( -5 \right)\centerdot \left( -5 \right)\centerdot \left( -5 \right)=\left( -5 \right)^{3}</math>,
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<math>\sqrt[3]{-125}</math>
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is defined as
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{{Displayed math||<math>\sqrt[3]{-125}=-5\,\textrm{.}</math>}}
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<math>\sqrt[3]{-125}=-5</math>
 
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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>.
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NOTE: As opposed to
 
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<math>\sqrt{-125}</math>
 
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(the square root of
 
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<math>-125</math>
 
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) which is not defined,
 
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<math>\sqrt[3]{-125}</math>
 
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is defined . In other words, there does not exist any number which satisfies
 
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<math>x^{\text{2}}=-\text{125}</math>, but there is a number
 
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<math>x</math>
 
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which satisfies
 
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<math>x^{\text{3}}=-\text{125}</math>.
 
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NOTE: It is possible to write the calculation in the solution as
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Note: It is possible to write the calculation in the solution as
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<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
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{{Displayed math||<math>\sqrt[3]{-125} = \sqrt[3]{(-5)^{3}} = (-5)^1 = -5\,,</math>}}
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<math>\begin{align}
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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
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& -5=\left( -125 \right)^{{1}/{3}\;}=\left( -125 \right)^{{2}/{6}\;}=\left( \left( -125 \right)^{2} \right)^{{1}/{6}\;} \\
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& =15625^{{1}/{6}\;}=5 \\
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{{Displayed math||<math>\begin{align}
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\end{align}</math>
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-5 = (-125)^{1/3} = (-125)^{2/6} = \bigl((-125)^2\bigr)^{1/6} = 15625^{1/6}=5\,\textrm{.}
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\end{align}</math>}}

Version vom 10:35, 30. Sep. 2008

Since \displaystyle -125 can be written as \displaystyle -125 = (-5)\cdot (-5)\cdot (-5) = (-5)^3, the number \displaystyle \sqrt[3]{-125} is defined as

Vorlage:Displayed math


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


Note: It is possible to write the calculation in the solution as

Vorlage:Displayed 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

Vorlage:Displayed math

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