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If we write $\sqrt{5}+\frac{1}{\sqrt{5}} + \sqrt{7} + \frac{1}{\sqrt{7}}$ in the form $\dfrac{a\sqrt{5} + b\sqrt{7}}{c}$ such that $a$, $b$, and $c$ are positive integers and $c$ is as small as possible, then what is $a+b+c$?

 Jun 16, 2019
 #1
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If we write $\sqrt{5}+\frac{1}{\sqrt{5}} + \sqrt{7} + \frac{1}{\sqrt{7}}$ in the form $\dfrac{a\sqrt{5} + b\sqrt{7}}{c}$ such that $a$, $b$, and $c$ are positive integers and $c$ is as small as possible, then what is $a+b+c$?

laugh

 Jun 16, 2019
 #2
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\(\phantom{= }\sqrt 5 + \dfrac{1}{\sqrt5} + \sqrt 7 + \dfrac{1}{\sqrt7}\\ = \dfrac{6\sqrt5}{5} + \dfrac{8\sqrt7}{7}\\ = \dfrac{42\sqrt5}{35} + \dfrac{40\sqrt7}{35}\\ = \dfrac{42\sqrt5 + 40\sqrt7}{35}\\ a + b + c = 42 + 40 + 35 = 117\)

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 Jun 22, 2019

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