Problem
ALG-B1-M03-P013 A Reciprocal System
#13
★★★☆☆ Level 3 of 5
Let \(x\ne0\) and \(x+\frac1x=3\). Find \(x^3+\frac1{x^3}\).
Use \(u^3\), where \(u=x+\frac1x\).
Let \(u=x+\frac1x=3\). Then \(u^3=x^3+\frac1{x^3}+3\left(x+\frac1x\right)\).
Thus \(27=x^3+\frac1{x^3}+9\), so \(x^3+\frac1{x^3}=18\).
A good continuation of reciprocal substitution.