dragondash2007 dragondash2007
  • 01-10-2020
  • Mathematics
contestada

find r^3+1/r^3 if r+1/r=sqrt2

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LammettHash
LammettHash LammettHash
  • 01-10-2020

Recall that

[tex](a+b)^3=a^3+3a^2b+3ab^2+b^3[/tex]

which means

[tex]\left(r+\dfrac1r\right)^3=r^3+3r^2\left(\dfrac1r\right)+3r\left(\dfrac1r\right)^2+\left(\dfrac1r\right)^3=r^3+3r+\dfrac3r+\dfrac1{r^3}[/tex]

Given that [tex]r+\frac1r=\sqrt2[/tex], we have

[tex](\sqrt2)^3=r^3+3\sqrt2+\dfrac1{r^3}[/tex]

[tex]\implies r^3+\dfrac1{r^3}=2\sqrt2-3\sqrt2=\boxed{-\sqrt2}[/tex]

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