Rationalizing this fraction involving square roots

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Homework Help Overview

The discussion revolves around simplifying a fraction involving square roots, specifically the expression $$\frac{\left( \sqrt{5}+\sqrt{3} \right) \left(\sqrt{5} + \sqrt{2} \right)}{\sqrt{5} + \sqrt{3} + \sqrt{2}}$$. Participants are exploring methods for rationalizing the denominator and simplifying the expression.

Discussion Character

  • Exploratory, Mathematical reasoning, Problem interpretation

Approaches and Questions Raised

  • One participant describes a brute force method involving multiplication by a conjugate to rationalize the denominator and expresses a desire for a more efficient technique. Another participant suggests an alternative approach that appears to complicate the process further. A third participant introduces a mathematical identity related to the square roots, proposing a potential simplification.

Discussion Status

The discussion is ongoing, with various methods being proposed and explored. While some participants offer insights and alternative approaches, there is no clear consensus on the most effective method for simplification. The conversation reflects a mix of attempts and suggestions without definitive conclusions.

Contextual Notes

Participants are navigating the complexities of rationalizing expressions with square roots, indicating a potential challenge in finding a straightforward solution. The original poster's inquiry about tricks to simplify the calculations suggests a desire to avoid lengthy computations.

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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Likes   Reactions: songoku, Steve4Physics and topsquark
Thank you very much for the help and explanation topsquark, PeroK
 

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