# Rationalizing this fraction involving square roots

• songoku

#### songoku

Homework Statement
Simplify:

$$\frac{\left( \sqrt{5}+\sqrt{3} \right) \left(\sqrt{5} + \sqrt{2} \right)}{\sqrt{5} + \sqrt{3} + \sqrt{2}}$$
Relevant Equations
Rationalization
I can do the question using brute force. First I multiply both the numerator and denominator by ##\sqrt{5} + \sqrt{3} - \sqrt{2}## then I simplify everything and rationalize again until no more square root in the denominator.

I want to ask if there is a trick to reduce the monstrous calculation

Thanks

Homework Statement:: Simplify:

$$\frac{\left( \sqrt{5}+\sqrt{3} \right) \left(\sqrt{5} + \sqrt{2} \right)}{\sqrt{5} + \sqrt{3} + \sqrt{2}}$$
Relevant Equations:: Rationalization

I can do the question using brute force. First I multiply both the numerator and denominator by ##\sqrt{5} + \sqrt{3} - \sqrt{2}## then I simplify everything and rationalize again until no more square root in the denominator.

I want to ask if there is a trick to reduce the monstrous calculation

Thanks
I believe you are stuck with that method. The alternate is probably even worse:
##( \sqrt{5} + \sqrt{3} + \sqrt{2} ) ( - \sqrt{5} + \sqrt{3} + \sqrt{2} ) ( \sqrt{5} - \sqrt{3} + \sqrt{2} ) ( \sqrt{5} + \sqrt{3} - \sqrt{2} ) = 24##

-Dan

• songoku and PeroK
If ##c = a - b##, then:
$$(\sqrt a + \sqrt b+ \sqrt c)^2 = 2(a + \sqrt{ab} + (\sqrt a + \sqrt b)\sqrt c) =2(\sqrt a + \sqrt b)(\sqrt a + \sqrt c)$$Hence:
$$\frac{(\sqrt a + \sqrt b)(\sqrt a + \sqrt c)}{\sqrt a + \sqrt b+ \sqrt c} = \frac{\sqrt a + \sqrt b+ \sqrt c}{2}$$

• • songoku, Steve4Physics and topsquark
Thank you very much for the help and explanation topsquark, PeroK