Exercise 10, § 7

Now prove that the group defined by the presentation

is reduced to the identity element. (The first relation entails $x^2y^8x^{-2} = y^{18}$ and $x^3y^8x^{-3} = y^{27}$; use the second relation to deduce that $y^{18} = y^{27}$, whence $y^9 = e$; the fact that $y^2$ is conjugate of $y^3$ then entails $y = e$, whence $x = e$.)