Finding the rationalizing factor of real numbers with denominator $1$

Question

I have a question which I could not solve after hours of research. It goes like this:

Find the rationalizing factor of $$\sqrt[3]{16} - \sqrt[3]{4} + 1$$

I can rationalize the denominator but can’t do questions like this. Please help me to solve this question and also other questions like this.

Thanks

Answer 1

Let $u = \sqrt[3]{4}$. Then we have to find the rationalising factor of $u^2 - u + 1$.

However, we know that $(u^2-u+1)(u+1) = u^3+1$. Therefore the rationalising factor is just $u+1$ or $ \sqrt[3]{4}+1$.