Question

Prove that $\sqrt[3]{33}$ is irrational.

Answer 1

Let $\sqrt[3]{33}$ be rational = $\frac{p}{q}$, where p and q belong to Z and p, q have no common factor except 1 also q > 1.

$\frac{p}{q}$ = $\sqrt[3]{33}$

Cubing both sides

$\frac{p^3}{q^3}=3$

Multiply both sides by $q^2$.

$\frac{p^3}{q}$ = $3q^2$, Clearly L.H.S is rational since p, q have no common factor.

$p^3,q$ also have no common factor while R.H.S is an integer.

L.H.S not equal to R.H.S which contradicts our assumption that $\sqrt[3]{33}$ is rational.