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Reading "Art of Problem Solving : Vol. 1". Stuck on Excercise 1-6 :

$$ {\sqrt2\over \sqrt6-2} $$

I know, we must rationalize, multiplying by

$$ { \sqrt6+2\over \sqrt6+2} $$

However, what would the final product be? Thanks!

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    $\begingroup$ First step: Work out through FOIL the product $(\sqrt{6}-2)(\sqrt{6}+2)$. What do you get? $\endgroup$ – imranfat Jul 21 '16 at 17:36
  • $\begingroup$ Do you know multiplication of surds?? $\endgroup$ – SchrodingersCat Jul 21 '16 at 17:37
  • $\begingroup$ If you know what to do, then do the thing you're supposed to do and you'll get to the final answer. $\endgroup$ – arctic tern Jul 21 '16 at 18:21
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specifically would be:

$$\frac{\sqrt{2}}{\sqrt{6} - 2} \cdot 1 = \frac{\sqrt{2}}{\sqrt{6} - 2} \cdot \frac{\sqrt{6} + 2}{\sqrt{6} + 2} = \frac{\sqrt{2}\sqrt{6} + 2\sqrt{2}}{6 + 2\sqrt{6} - 2\sqrt{6} - 4}$$

in these last term remember that $(a + b)\cdot(a - b) = a² - ab + ab - b²$ for this reason in the denominator $({\sqrt{6} - 2}) \cdot ({\sqrt{6} + 2}) = \sqrt{6}² - 2\sqrt{6} + 2\sqrt{6} - 4 = 6 - 4 = 2$

So,

$$\frac{\sqrt{2}\sqrt{6} + 2\sqrt{2}}{6 + 2\sqrt{6} - 2\sqrt{6} - 4} = \frac{\sqrt{2\cdot6} + 2\sqrt{2}}{2} = \frac{\sqrt{12}}{2} + \frac{2\sqrt{2}}{2} = \frac{\sqrt{4\cdot3}}{2} + \sqrt{2} = \frac{\sqrt{4}\sqrt{3}}{2} + \sqrt{2} = \sqrt{3} + \sqrt{2}$$

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  • $\begingroup$ Thank you! I know understand. $\endgroup$ – DarkRunner Jul 21 '16 at 18:41
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$$\frac{\sqrt2}{\sqrt6-2}\cdot\frac{\sqrt6+2}{\sqrt6+2}=\frac{2\sqrt3+2\sqrt2}{2}=\sqrt3+\sqrt2$$

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