Rationalising the Surds

Question

Please help me rationalise and simplify: $$ \frac{1}{\sqrt[3]{2} - 1} \ - \ \frac{2}{\sqrt{3} - 2} \ . $$

I have tried using the cube of the denominator and the square of the denominator on the second one but it stuffs up when I try to simplify.

Thanks.

Answer 1

Hint : $(2^{1/3}-1)(1+2^{1/3}+2^{2/3})=(2-1)$

$(\sqrt3 - 2)(\sqrt3 +2)=(3-2^2) $

Answer 2

The hints provided by Awesome are sufficient to get the answer immediately. In general if $\alpha$ is a root of an irreducible polynomial $f(x)$ with rational coefficients (in your case $x^3-2=0$), and you are asked to find the value of something like $ \frac{1}{\alpha^2+a\alpha +b}$, or inverse of a general polynomial expression $g(\alpha)$, carry out Euclidean algorithm for gcd with $f(x)$ and $g(x)$ (same procedure in Bezout's theorem with integers) which will give an identity $a(x)f(x) +b(x) g(x)\equiv 1$.

Substituting $\alpha$ in the identity first term evaluates to zero and we get that $b(\alpha)$ is the inverse of $g(\alpha)$ we are looking for. Here $b(\alpha)$ being a polynomial expression there are are no denominators.