# Fractional Surds - Simplifying and Rational Denominator

Simplify $\frac{3}{\sqrt5 + 2} - \frac{\sqrt2}{2.\sqrt2 - 1}$, writing your answer with a rational denominator.

So i have solved questions like this in my whole life. But i'm just confused can someone PLEASE solve it for me.?

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Please solve it :| –  Trent Mar 18 '13 at 10:07
multiply it by conjugate ... so that you can square both terms. –  Santosh Linkha Mar 18 '13 at 10:09
I've done about 5 methods, including yours . BUT IT JUST WONT WORK WHERE AM I GOING WRONG? it's really annoying... –  Trent Mar 18 '13 at 10:11
Write up your failed attempt. Then we can tell you where you are going wrong! I don't have ESP. –  jim Mar 18 '13 at 10:13
it's really hard writting in that format. with the website –  Trent Mar 18 '13 at 10:13

$$\frac{3}{\sqrt5 + 2} - \frac{\sqrt2}{2.\sqrt2 - 1}$$ $$=\frac{3.(\sqrt5 - 2)}{(\sqrt5 + 2).(\sqrt5 - 2)} - \frac{\sqrt2.(2.\sqrt2 + 1)}{(2.\sqrt2 - 1).(2.\sqrt2 + 1)}$$ $$=\frac{3.(\sqrt5 - 2)}{\sqrt5^2 -2^2} - \frac{\sqrt2.(2.\sqrt2 + 1)}{(2.\sqrt2)^2 - 1^2}$$ $$=\frac{3.(\sqrt5 - 2)}{1} - \frac{\sqrt2.(2.\sqrt2 + 1)}{(2.\sqrt2)^2 - 1^2}$$ $$=\frac{3.(\sqrt5 - 2)}{1} - \frac{\sqrt2.(2.\sqrt2 + 1)}{7}$$ $$=\frac{21.(\sqrt5 - 2) - \sqrt2.(2.\sqrt2 + 1)}{7}$$ $$=\frac{21.\sqrt5 - 42 - 2.2 -\sqrt2}{7}$$ $$=\frac{21.\sqrt5 - 46 -\sqrt2}{7}$$