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Compute the number \(\left( \frac{r}{s} \right)^3\) if \(r\) and \(s\) are non-zero numbers satisfying \(\frac{1}{r + s} = \frac{1}{r} + \frac{1}{s}.\)
 

 May 20, 2019
 #1
avatar+6046 
+3

\(\dfrac{1}{r+s}=\dfrac 1 r + \dfrac 1 s\\~\\ \dfrac{\frac 1 s}{\frac r s + 1}= \dfrac 1 s\left( \dfrac{1}{\frac r s}+1\right)\\~\\ \dfrac{1}{\frac r s + 1}= \dfrac{1}{\frac r s}+1 \)

 

\(u=\dfrac r s\\~\\ \dfrac{1}{1+u}= \dfrac 1 u + 1\\~\\ \dfrac{1}{1+u} = \dfrac{1+u}{u}\\~\\ u = u^2+2u+1\\~\\ u^2 + u + 1 = 0\\~\\ u = \dfrac{-1\pm \sqrt{1-4}}{2} = -\dfrac 1 2\pm \dfrac{\sqrt{3}}{2}i = \\~\\ e^{i2\pi/3},~e^{i4\pi/3}\\~\\ u^3 = e^{i6\pi/3},~e^{i12\pi/3} =1,1\\~\\ \dfrac r s = 1\)

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 May 20, 2019
 #2
avatar+166 
0

Thanks

 May 25, 2019

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