Express \(\sqrt 5\) by rational numbers and a number \((\sqrt 5 + \sqrt 7)\) using addition, subtraction, multiplication and division.

Edited January 2Jan 2 by Genady

[math]\big(\sqrt{7} + \sqrt{5}\big)\big(\sqrt{7} - \sqrt{5}\big) = 2
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\big(\sqrt{7} - \sqrt{5}\big) = \dfrac{2}{\big(\sqrt{7} + \sqrt{5}\big)}
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\sqrt{5} = \dfrac{1}{2}\Big(\big(\sqrt{7} + \sqrt{5}\big) - \big(\sqrt{7} - \sqrt{5}\big)\Big)
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= \dfrac{1}{2}\Big(\big(\sqrt{7} + \sqrt{5}\big) - \dfrac{2}{\big(\sqrt{7} + \sqrt{5}\big)}\Big)
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= \dfrac{\big(\sqrt{7} + \sqrt{5}\big)}{2} - \dfrac{1}{\big(\sqrt{7} + \sqrt{5}\big)}[/math]

 

12 minutes ago, KJW said:

(7–√+5–√)(7–√−5–√)=2(7–√−5–√)=2(7–√+5–√)5–√=12((7–√+5–√)−(7–√−5–√))=12((7–√+5–√)−2(7–√+5–√))=(7–√+5–√)2−1(7–√+5–√)

 

This is a good one. Nicer than what I had in mind.