Rationalizing this fraction involving square roots

AI Thread Summary
The discussion revolves around simplifying the fraction involving square roots, specifically the expression (√5 + √3)(√5 + √2) / (√5 + √3 + √2). One participant describes using brute force by multiplying the numerator and denominator by (√5 + √3 - √2) to rationalize the denominator. Another contributor suggests that this method may be cumbersome and proposes an alternative approach involving algebraic manipulation. Ultimately, the conversation highlights the complexity of rationalizing such expressions while seeking more efficient methods. The participants appreciate the insights shared during the discussion.
songoku
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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
 
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songoku said:
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
 
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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:
$$\frac{(\sqrt a + \sqrt b)(\sqrt a + \sqrt c)}{\sqrt a + \sqrt b+ \sqrt c} = \frac{\sqrt a + \sqrt b+ \sqrt c}{2}$$
 
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Thank you very much for the help and explanation topsquark, PeroK
 
I tried to combine those 2 formulas but it didn't work. I tried using another case where there are 2 red balls and 2 blue balls only so when combining the formula I got ##\frac{(4-1)!}{2!2!}=\frac{3}{2}## which does not make sense. Is there any formula to calculate cyclic permutation of identical objects or I have to do it by listing all the possibilities? Thanks
Since ##px^9+q## is the factor, then ##x^9=\frac{-q}{p}## will be one of the roots. Let ##f(x)=27x^{18}+bx^9+70##, then: $$27\left(\frac{-q}{p}\right)^2+b\left(\frac{-q}{p}\right)+70=0$$ $$b=27 \frac{q}{p}+70 \frac{p}{q}$$ $$b=\frac{27q^2+70p^2}{pq}$$ From this expression, it looks like there is no greatest value of ##b## because increasing the value of ##p## and ##q## will also increase the value of ##b##. How to find the greatest value of ##b##? Thanks
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