We use cookies to personalise content and advertisements and to analyse access to our website. Furthermore, our partners for online advertising receive pseudonymised information about your use of our website. cookie policy and privacy policy.
 
+0  
 
+2
63
2
avatar+93 

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+5172 
+3

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

 

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

.
 May 20, 2019
 #2
avatar+93 
0

Thanks

 May 25, 2019

13 Online Users

avatar
avatar
avatar