Solution 3.1:7b
From Förberedande kurs i matematik 1
m (Lösning 3.1:7b moved to Solution 3.1:7b: Robot: moved page) |
m |
||
(One intermediate revision not shown.) | |||
Line 1: | Line 1: | ||
- | {{ | + | We multiply the top and bottom of the fraction by the conjugate of the denominator, |
- | < | + | <math>\sqrt{7}+\sqrt{5}</math>, and see what it leads to, |
- | {{ | + | |
+ | {{Displayed math||<math>\begin{align} | ||
+ | \frac{5\sqrt{7}-7\sqrt{5}}{\sqrt{7}-\sqrt{5}} | ||
+ | &= \frac{5\sqrt{7}-7\sqrt{5}}{\sqrt{7}-\sqrt{5}}\cdot \frac{\sqrt{7}+\sqrt{5}}{\sqrt{7}+\sqrt{5}}\\[10pt] | ||
+ | &= \frac{(5\sqrt{7}-7\sqrt{5})(\sqrt{7}+\sqrt{5})}{(\sqrt{7})^{2}-(\sqrt{5})^{2}}\\[10pt] | ||
+ | &= \frac{5\sqrt{7}\cdot\sqrt{7}+5\sqrt{5}\cdot\sqrt{7}-7\sqrt{5}\cdot\sqrt{7}-7\sqrt{5}\cdot\sqrt{5}}{7-5}\\[10pt] | ||
+ | &= \frac{5(\sqrt{7})^{2}+5\sqrt{5}\sqrt{7}-7\sqrt{5}\sqrt{7}-7(\sqrt{5})^{2}}{2}\\[10pt] | ||
+ | &= \frac{5\cdot 7+5\sqrt{5}\sqrt{7}-7\sqrt{5}\sqrt{7}-7\cdot 5}{2}\\[10pt] | ||
+ | &= \frac{5\sqrt{5}\sqrt{7}-7\sqrt{5}\sqrt{7}}{2}\\[10pt] | ||
+ | &= \frac{(5-7)\sqrt{5}\sqrt{7}}{2}\\[10pt] | ||
+ | &= \frac{-2\sqrt{5\cdot 7}}{2}\\[10pt] | ||
+ | &= -\sqrt{35}\,\textrm{.} | ||
+ | \end{align}</math>}} |
Current revision
We multiply the top and bottom of the fraction by the conjugate of the denominator, \displaystyle \sqrt{7}+\sqrt{5}, and see what it leads to,
\displaystyle \begin{align}
\frac{5\sqrt{7}-7\sqrt{5}}{\sqrt{7}-\sqrt{5}} &= \frac{5\sqrt{7}-7\sqrt{5}}{\sqrt{7}-\sqrt{5}}\cdot \frac{\sqrt{7}+\sqrt{5}}{\sqrt{7}+\sqrt{5}}\\[10pt] &= \frac{(5\sqrt{7}-7\sqrt{5})(\sqrt{7}+\sqrt{5})}{(\sqrt{7})^{2}-(\sqrt{5})^{2}}\\[10pt] &= \frac{5\sqrt{7}\cdot\sqrt{7}+5\sqrt{5}\cdot\sqrt{7}-7\sqrt{5}\cdot\sqrt{7}-7\sqrt{5}\cdot\sqrt{5}}{7-5}\\[10pt] &= \frac{5(\sqrt{7})^{2}+5\sqrt{5}\sqrt{7}-7\sqrt{5}\sqrt{7}-7(\sqrt{5})^{2}}{2}\\[10pt] &= \frac{5\cdot 7+5\sqrt{5}\sqrt{7}-7\sqrt{5}\sqrt{7}-7\cdot 5}{2}\\[10pt] &= \frac{5\sqrt{5}\sqrt{7}-7\sqrt{5}\sqrt{7}}{2}\\[10pt] &= \frac{(5-7)\sqrt{5}\sqrt{7}}{2}\\[10pt] &= \frac{-2\sqrt{5\cdot 7}}{2}\\[10pt] &= -\sqrt{35}\,\textrm{.} \end{align} |