If (r+1/r)^2 = 5, what is the value of (r^3+1/r^3)^2?

\({\left( {r + {1 \over r}} \right)^2} = 5\,\,\,\,\,\left( * \right)\)

\(? = \,{\left( {{r^3} + {1 \over {{r^3}}}} \right)^2}\)

\(?\,\,\, = \,\,\,{\left[ {\left( {r + {1 \over r}} \right)\left( {{r^2} - r \cdot {1 \over r} + {1 \over {{r^2}}}} \right)} \right]^{\,2}} = {\left( {r + {1 \over r}} \right)^2}{\left( {{r^2} - 1 + {1 \over {{r^2}}}} \right)^2}\,\,\,\mathop = \limits^{\left( * \right)} \,\,\,5 \cdot {\left( {{r^2} - 1 + {1 \over {{r^2}}}} \right)^2}\,\,\,\mathop = \limits^{\left( {**} \right)} \,\,\,20\)

\(\left( {**} \right)\,\,{r^2} - 1 + {1 \over {{r^2}}} = \left( {{r^2} + 2 + {1 \over {{r^2}}}} \right) - 3 = {\left( {r + {1 \over r}} \right)^2} - 3\,\,\,\mathop = \limits^{\left( * \right)} \,\,\,2\)


The correct answer is (E).


We follow the notations and rationale taught in the GMATH method.

Regards,
Fabio.