Solution 3.1:6b
From Förberedande kurs i matematik 1
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- | {{ | + | The root sign in the denominator lies in a quadratic term and we therefore expand first the quadratic |
- | + | ||
- | {{ | + | {{Displayed math||<math>\begin{align} |
+ | (\sqrt{3}-2)^2 | ||
+ | &= (\sqrt{3}\,)^{2} - 2\cdot\sqrt{3}\cdot 2 + 2^{2}\\[5pt] | ||
+ | &= 3-4\sqrt{3}+4\\[5pt] | ||
+ | &= 7-4\sqrt{3}\,\textrm{.} | ||
+ | \end{align}</math>}} | ||
+ | |||
+ | Thus, | ||
+ | |||
+ | {{Displayed math||<math>\frac{1}{(\sqrt{3}-2)^{2}-2} = \frac{1}{7-4\sqrt{3}-2} = \frac{1}{5-4\sqrt{3}}</math>}} | ||
+ | |||
+ | 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 <math>5+4\sqrt{3}</math>, | ||
+ | |||
+ | {{Displayed math||<math>\begin{align} | ||
+ | \frac{1}{5-4\sqrt{3}} | ||
+ | &= \frac{1}{5-4\sqrt{3}}\cdot \frac{5+4\sqrt{3}}{5+4\sqrt{3}}\\[5pt] | ||
+ | &= \frac{5+4\sqrt{3}}{5^{2}-(4\sqrt{3})^{2}}\\[5pt] | ||
+ | &= \frac{5+4\sqrt{3}}{5^{2}-4^{2}(\sqrt{3})^{2}}\\[5pt] | ||
+ | &= \frac{5+4\sqrt{3}}{25-16\cdot 3}\\[5pt] | ||
+ | &= \frac{5+4\sqrt{3}}{-23}\\[5pt] | ||
+ | &= -\frac{5+4\sqrt{3}}{23}\,\textrm{.} | ||
+ | \end{align}</math>}} |
Current revision
The root sign in the denominator lies in a quadratic term and we therefore expand first the quadratic
\displaystyle \begin{align}
(\sqrt{3}-2)^2 &= (\sqrt{3}\,)^{2} - 2\cdot\sqrt{3}\cdot 2 + 2^{2}\\[5pt] &= 3-4\sqrt{3}+4\\[5pt] &= 7-4\sqrt{3}\,\textrm{.} \end{align} |
Thus,
\displaystyle \frac{1}{(\sqrt{3}-2)^{2}-2} = \frac{1}{7-4\sqrt{3}-2} = \frac{1}{5-4\sqrt{3}} |
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}}\cdot \frac{5+4\sqrt{3}}{5+4\sqrt{3}}\\[5pt] &= \frac{5+4\sqrt{3}}{5^{2}-(4\sqrt{3})^{2}}\\[5pt] &= \frac{5+4\sqrt{3}}{5^{2}-4^{2}(\sqrt{3})^{2}}\\[5pt] &= \frac{5+4\sqrt{3}}{25-16\cdot 3}\\[5pt] &= \frac{5+4\sqrt{3}}{-23}\\[5pt] &= -\frac{5+4\sqrt{3}}{23}\,\textrm{.} \end{align} |