Question

Find the factor which will rationalise:
$\sqrt[3]{33}-1$

Answer 1

We need to find the factor which when multiplied by $(\sqrt[3]{33}-1)$ will lead to a rational number

Let $x=3^{\frac{1}{3}}$ and $y=1$, then $x^3$ and $y^3$ are both rational

Also, $x^3-y^3=(x-y)(x^2+xy+y^2)$

Hence, $x^2+xy+y^2$ is the required factor.

Substituting the value for $x$ and $y$, then the required factor is

$3^{\frac{2}{3}}+3^{\frac{1}{3}}+1$