Solution 3.1:6b

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m (Lösning 3.1:6b moved to Solution 3.1:6b: Robot: moved page)
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The root sign in the denominator lies in a quadratic term and we therefore expand first the quadratic
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<center> [[Image:3_1_6b.gif]] </center>
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<math>\left( \sqrt{3}-2 \right)^{2}</math>
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using the square rule
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<math>\begin{align}
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& \left( \sqrt{3}-2 \right)^{2}=\left( \sqrt{3} \right)^{2}-2\centerdot \sqrt{3}\centerdot 2+2^{2} \\
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& =3-4\sqrt{3}+4=7-4\sqrt{3} \\
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\end{align}</math>
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Thus,
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<math>\begin{align}
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& \left( \sqrt{3}-2 \right)^{2}=\left( \sqrt{3} \right)^{2}-2\centerdot \sqrt{3}\centerdot 2+2^{2} \\
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& =3-4\sqrt{3}+4=7-4\sqrt{3} \\
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& \frac{1}{\left( \sqrt{3}-2 \right)^{2}-2}=\frac{1}{7-4\sqrt{3}-2}=\frac{1}{5-4\sqrt{3}} \\
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\end{align}</math>
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and in the expression we can get rid of the root sign from the denominator by multiplying the top and bottom of the equation by the conjugate
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<math>5+4\sqrt{3}</math>,
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<math>\begin{align}
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& \frac{1}{5-4\sqrt{3}}=\frac{1}{5-4\sqrt{3}}\centerdot \frac{5+4\sqrt{3}}{5+4\sqrt{3}}=\frac{5+4\sqrt{3}}{5^{2}-\left( 4\sqrt{3} \right)^{2}} \\
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& =\frac{5+4\sqrt{3}}{5^{2}-4^{2}\left( \sqrt{3} \right)^{2}}=\frac{5+4\sqrt{3}}{25-16\centerdot 3} \\
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& =\frac{5+4\sqrt{3}}{-23}=-\frac{5+4\sqrt{3}}{23} \\
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\end{align}</math>

Revision as of 09:00, 23 September 2008

The root sign in the denominator lies in a quadratic term and we therefore expand first the quadratic \displaystyle \left( \sqrt{3}-2 \right)^{2} using the square rule


\displaystyle \begin{align} & \left( \sqrt{3}-2 \right)^{2}=\left( \sqrt{3} \right)^{2}-2\centerdot \sqrt{3}\centerdot 2+2^{2} \\ & =3-4\sqrt{3}+4=7-4\sqrt{3} \\ \end{align}

Thus,


\displaystyle \begin{align} & \left( \sqrt{3}-2 \right)^{2}=\left( \sqrt{3} \right)^{2}-2\centerdot \sqrt{3}\centerdot 2+2^{2} \\ & =3-4\sqrt{3}+4=7-4\sqrt{3} \\ & \frac{1}{\left( \sqrt{3}-2 \right)^{2}-2}=\frac{1}{7-4\sqrt{3}-2}=\frac{1}{5-4\sqrt{3}} \\ \end{align}

and in the expression we can get rid of the root sign from the denominator by multiplying the top and bottom of the equation by the conjugate \displaystyle 5+4\sqrt{3},


\displaystyle \begin{align} & \frac{1}{5-4\sqrt{3}}=\frac{1}{5-4\sqrt{3}}\centerdot \frac{5+4\sqrt{3}}{5+4\sqrt{3}}=\frac{5+4\sqrt{3}}{5^{2}-\left( 4\sqrt{3} \right)^{2}} \\ & =\frac{5+4\sqrt{3}}{5^{2}-4^{2}\left( \sqrt{3} \right)^{2}}=\frac{5+4\sqrt{3}}{25-16\centerdot 3} \\ & =\frac{5+4\sqrt{3}}{-23}=-\frac{5+4\sqrt{3}}{23} \\ \end{align}

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